GNSS signal processing methods and apparatus with ambiguity convergence indication

ABSTRACT

Methods and apparatus are provided for estimating parameters, i.e. ambiguities, derived from GNSS signals. Observations of a GNSS signal from each of a plurality of GNSS satellites are obtained ( 2120 ). The observations are fed to a filter having a state vector comprising a float ambiguity for each received frequency of the GNSS signals ( 2140 ). The filter estimates a float value for each float ambiguity of the state vector and co-variance values associated with the state vector. Integer values are assigned to at least a subgroup of the estimated float values to define a plurality of integer ambiguity candidate sets ( 2160 ). A weighted average of the candidate sets is formed ( 2200 ). A formal precision value based on covariance values of the filter is determined ( 2205 ), the formal precision value being a measure for an achievable precision. An achieved precision value of the weighted average is determined ( 2210 ). The achieved precision value is compared with the formal precision value to obtain a convergence value ( 2215 ). A convergence of the determination of the state vector is indicated ( 2218 ). Ambiguities of the weighted average can be used in subsequent operations to aid in determining a position of the receiver or can be used to prepare data, e.g., in a network processor that can be used to augment position information of a rover.

RELATED APPLICATIONS

This application claims benefit of U.S. Provisional Application forPatent 61/189,382 filed Aug. 19, 2008, the content of which isincorporated herein by this reference.

This application Ser. No. 13/059,416 is related to the following U.S.national stage patent applications which also claim benefit of U.S.Provisional Application for Patent 61/189,382: 13/059,463 filed Feb. 17,2011 (§371); 13/059,415 filed Feb. 16, 2011 (§371); 13/059,413 filedFeb. 16, 2011 (§371); and 13/059,428 filed Feb. 17, 2011 (§371).

FIELD OF THE INVENTION

The invention relates to position estimation methods and apparatuses,and especially to such methods and apparatuses based on navigationalsatellite system signals. The fields of application of the methods andapparatuses include, but are not limited to, navigation, map-making,land surveying, civil engineering, disaster prevention and relief, andscientific research.

BACKGROUND

1. Introduction

Global navigational satellite systems (GNSS) include Global PositioningSystem (GPS) (United States), GLONASS (Russia), Galileo (Europe) andCOMPASS (China) (systems in use or in development). A GNSS typicallyuses a plurality of satellites orbiting the earth. The plurality ofsatellites forms a constellation of satellites. A GNSS receiver detectsa code modulated on an electromagnetic signal broadcasted by asatellite. The code is also called a ranging code. Code detectionincludes comparing the bit sequence modulated on the broadcasted signalwith a receiver-side version of the code to be detected. Based on thedetection of the time of arrival of the code for each of a series of thesatellites, the GNSS receiver estimates its position. Positioningincludes geolocation, i.e. the positioning on the surface of the Earth.

An overview of GPS, GLONASS and Galileo is provided in sections 2.1.1,2.1.2 and 2.1.3 of Sandra Verhagen, The GNSS integer ambiguities:estimation and validation, Delft University of Technology, 2004, ISBN90-804147-4-3 (herein referred as “[1]”) (which was also published inPublications on Geodesy 58, Delft, 2005, ISBN-13: 978 90 6132 290 0,ISBN-10: 90 6132 290).

Positioning using GNSS signal codes provides a limited accuracy, notablydue to the distortion the code is subject to upon transmission throughthe atmosphere. For instance, the GPS includes the transmission of acoarse/acquisition (C/A) code at 1575.45 MHz, the so-called L1frequency. This code is freely available to the public, in comparison tothe Precise (P) code, which is reserved for military applications. Theaccuracy of code-based positioning using the GPS C/A code isapproximately 15 meters, when taking into account both the electronicuncertainty associated with the detection of the C/A code (electronicdetection of the time of arrival of the pseudorandom code) and othererrors including those caused by ionospheric and tropospheric effects,ephemeris errors, satellite clock errors and multipath propagation.

An alternative to positioning based on the detection of a code ispositioning based on carrier phase measurements. In this alternativeapproach, the carrier phase of the GNSS signal transmitted from thesatellite is detected, not the code modulated on the signal transmittedfrom the satellite.

The approach based on carrier phase measurements has the potential toprovide much greater position precision, i.e. up to centimeter-level oreven millimeter-level precision, compared to the code-based approach.The reason may be intuitively understood as follows. The code, such asthe GPS C/A code on the L1 band, is much longer than one cycle of thecarrier on which the code is modulated. The position resolution maytherefore be viewed as greater for carrier phase detection than for codedetection.

However, in the process of estimating the position based on carrierphase measurements, the carrier phases are ambiguous by an unknownnumber of cycles (this is for instance explained in [1], section 1.1,second paragraph). The phase of a received signal can be determined, butthe cycle cannot be directly determined in an unambiguous manner. Thisis the so-called “integer ambiguity problem”, “integer ambiguityresolution problem” or “phase ambiguity resolution problem”.

GNSS observation equations for code observations and for phaseobservations are for instance provided in [1], respectively sections2.2.1 and 2.2.2. An introduction to the GNSS integer resolution problemis provided in [1], section 3. The idea of using carrier phase data forGNSS positioning was however already introduced in 1984 in Remondi,Using the Global Positioning System (GPS) Phase Observable for RelativeGeodesy: Modeling Processing and Results, Center for Space Research, TheUniversity of Texas at Austin, May, 1984 (herein referred as “[2]”).

The basic principles of the GNSS integer resolution problem will be nowexplained with reference to FIGS. 1 to 4. Further explanations are thenprovided, with mathematical support and explanations of the furtherfactors generally involved in implementing an integer resolution systemfor precise position estimation.

2. Basic Principles of the GNSS Integer Resolution Problem

FIG. 1 schematically illustrates a GNSS with only two satellites and onereceiver. A mobile receiver is also called a rover, a stationaryreceiver may be called a base station (useful for differentialprocessing, e.g., DGPS) or a reference station (useful for networkprocessing). While an actual GNSS involves more than two satellites, inorder to illustrate the basic principles of GNSS carrier-phasemeasurements, only two satellites are represented on FIG. 1. Eachsatellite broadcasts a signal (illustrated by the arrows originatingfrom the satellites and going in the direction of the receiver). Aportion of the carrier of each of the signals is schematicallyrepresented in the vicinity of the receiver. The portion of the carrieris represented in FIG. 1 in a non-modulated form for the sake ofclarity. In reality, a code is modulated, e.g. bi-phase shift key(BPSK)-modulated, on the caner.

At one point in time, the receiver can measure the phase of the carrierof the received signal. The receiver can also track the carrier phase,and lock onto it to track the additional cycles of the carrier due tothe changing distance between the receiver and the satellite. The aspectof tracking the phase in time is however disregarded for now, for thesake of clarity, but will be explained later.

While the receiver can measure the phase of the carrier of the receivedsignal, the number of cycles between the satellite and the receiver isunknown. The distance corresponding for instance to one cycle of a GPSL1 frequency carrier (1575.45 MHz) is about 19 centimeters (taking intoaccount the propagation speed of the radio signal, i.e. the speed oflight). The distance of 19 centimeters corresponds to the wavelength ofthe carrier. In other words, the carrier phase can be measured startingat one point in time, but the integer number of cycles from thesatellite to the receiver, e.g. at the starting point in time, isunknown. Also, the difference between the number of cycles from thefirst satellite to the receiver and the number of cycles from the secondsatellite to the receiver is unknown.

The amplitude peaks of some successive cycles of a GNSS signal carrierin FIG. 1 are illustrated by the indications “−1”, “0”, “+1”, “+2”. Thisintuitively illustrates the integer ambiguity as to the number ofcycles. The peak referred to by the indication “0” corresponds to aparticular number of cycles from the receiver to the satellite. The peakreferred to by the indication “+1” corresponds to one more cycle, thepeak referred to by the indication “−1” corresponds to one less cycle,and so on. The estimation of the position of the receiver depends on theknowledge of the correct number of cycles from the satellite to thereceiver.

A given number of cycles from the first satellite to the receiver (e.g.corresponding to the indication “+1” on the left-hand side of FIG. 1)and a given number of cycles from the second satellite to the receiver(e.g. corresponding to the indication “−1” on the right-hand side ofFIG. 1) leads to one estimated position. In other words, it can be seenthat the integer couple (+1, −1) leads to one estimated position in thetwo-dimensional illustration. Another combination of number of cycles toeach of the satellites leads to another estimated position. Forinstance, the combination (0, +1) leads to another estimated position inthe two-dimensional illustration. This will be better understood withreference to FIGS. 2 a to 4.

FIG. 2 a discloses a two-dimensional uncertainty region wherein thereceiver is known to be, or at least is known to be very likely to be.The knowledge that the receiver is within this uncertainty circle mayfor instance be based on a coarse code-based position estimation. Whilean actual GNSS implementation involves more than a two-dimensionaluncertainty region, the basic principles of carrier-phase measurementsmay be understood by referring to a two-dimensional space of unknownsonly. It is known or assumed that the position of the receiver is withinthe uncertainty region limited by the represented circle, but theposition within the circle is unknown yet.

FIG. 2 b discloses the two-dimensional uncertainty region of FIG. 2 awith, in addition, a dashed line with an arrow. The dashed linerepresents the direction towards one GNSS satellite, called Satellite 1.Satellite 1 broadcasts a signal containing a carrier. The parallel linesrepresented in the uncertainty region are the lines wherein the receivermay be located, when assuming that only the carrier of Satellite 1 isused for position estimation. Each line corresponds to a particular wavefront of the signal broadcasted by Satellite 1. Adjacent wave fronts areseparated by one carrier wavelength. Depending on the correct number ofcycles to Satellite 1, the receiver may be viewed as being located onone of these lines. Due to the integer ambiguity, the correct positionline is unknown.

FIG. 2 c represents the same elements as FIG. 2 b, i.e. the uncertaintyregion and the carrier wave fronts of the signal broadcasted bySatellite 1. In addition, FIG. 2 c includes a second dashed line with anarrow schematically indicating the direction of a second GNSS satellite,i.e. Satellite 2. The lines which are perpendicular to the direction ofSatellite 2 represent the carrier wave fronts of the signal broadcastedby Satellite 2. Depending on the correct number of cycles to Satellite2, the receiver may be viewed as being located on one of these lines.

The carrier of the signal broadcasted by Satellite 2 may be used forposition estimation in addition to the position information derived fromthe carrier associated with Satellite 1. The uncertainty can thereforebe reduced. Instead of being able to only assume that the position ofthe receiver is somewhere within the circle (of FIG. 2 a), or to onlyassume that the position of the receiver is on one of the straight linesrepresented on FIG. 2 b, it can now be assumed that the position of thereceiver is on one of the lines corresponding to the wave fronts ofSatellite 1 and simultaneously on one of the parallel linescorresponding to Satellite 2. In other words, by using carrier phasemeasurements from the two satellites, it can be assumed that theposition of the receiver is or is very likely to be on the intersectionbetween a line associated with a wave front of Satellite 1 and a lineassociated with a wave front of Satellite 2.

It can be seen that there are 28 line intersections within the exemplaryuncertainty region of FIG. 2 c. These intersections correspond to 28possible positions for the receiver. The position estimation problemwithin the uncertainty region is therefore reduced to the problem offinding on which one of these intersections the receiver is located.

More than two satellites are available to estimate the receiverposition. The carrier of a signal from a further satellite can thereforebe used in an attempt to resolve the integer ambiguity.

FIG. 2 d represents the same elements as FIG. 2 c, i.e. the uncertaintyregion and the carrier wave fronts of the signals broadcasted bySatellite 1 and by Satellite 2. In addition, FIG. 2 d includes a thirddashed line with an arrow indicating the direction of a third GNSSsatellite, i.e. Satellite 3. Accordingly, a third set of carrier wavefronts, associated with Satellite 3, are represented.

The third set of wave fronts added on FIG. 2 d and superposed on thepattern with the exemplary 28 intersections of FIG. 2 c providesadditional information to assist in resolving the integer ambiguity. Howwell a wave front associated with Satellite 3 fits with an intersectionbetween the wave fronts associated with Satellite 1 and Satellite 2provides an indication as to the probability that one particularintersection is the correct position. For instance, the triplet (0, 0,0) can be intuitively regarded as providing a highly likely combinationof wave fronts. The receiver is likely to be located at the intersectioncorresponding to the triplets (0, 0, 0) of wave fronts. This is howevernot the only possible intersection, also called a node.

In order to assign to each intersection of wave fronts associated withSatellite 1 and Satellite 2 (as illustrated on FIG. 2 c) a probabilityof being the correct position, each intersection may be considered oneby one. FIG. 2 e illustrates a particular intersection, or node, beingconsidered, or searched. The particular intersection, or search nodelocation (using the label of FIG. 2 e), is (+2, 0), i.e. wave front “+2”associated with Satellite 1 with wave front “0” associated withSatellite 2. How far the search node is from the closest wave frontassociated with Satellite 3 may intuitively be viewed as providing anindication of the probability that the search node corresponds to thecorrect position.

The wave front “+2” associated with Satellite 3 is the closest wavefront from the search node (+2, 0). The wave front “+2” associated withSatellite 3 is however relatively far from the search node, i.e. thesuperposition match is not good. It results that the triplet (+2, 0, +2)is relatively unlikely to correspond to the correct position.

Each combination of a wave front associated with Satellite 1 and a wavefront associated with Satellite 2 may be considered and a probabilitymay be assigned to each one of these integer combinations, also called“ambiguities” or “integer ambiguities”. This is shown in FIG. 3. Thevertical arrow which is associated with each intersection, or searchnode, gives an indication of the probability that the search nodecorresponds to the correct position. Note that the probabilities shownon FIG. 3 are exemplary and do not necessarily correspond to thesituation illustrated in FIGS. 2 d and 2 e.

FIG. 3 shows that the couple (0, 0) has a predominant probability withinthe uncertainty region. It is highly likely that this couple of integerscorresponds to the correct integer solution from which the receiverposition can be most precisely derived.

FIG. 4 shows another exemplary probability map wherein the probabilityassociated with each intersection of wave fronts is indicated by avertical arrow. However, in FIG. 4, two search nodes, corresponding tothe couples (0, 0) and (+1, −1), have an almost equal probability. Inthis situation, it is highly likely that one of these two integercouples corresponds to the correct integer solution from which thereceiver position can be most precisely derived.

The GNSS integer resolution problem is therefore the problem consistingin estimating with the highest possible confidence the correct set ofinteger values for the ambiguities, so as to provide high precision GNSSpositioning. It may include scanning through possible integercombinations or search nodes (as illustrated in FIG. 2 e) usingstatistical measures to assess which combination is the correct one.

3. Mathematical Formulation and Further Considerations

3.1 Observation Equations

The linear GNSS observation equations are provided byy=Aa+Bb+e  (1)wherein

-   -   y is the GPS observation vector of order m (the vector of        observables),    -   a and b are the unknown parameter vectors of dimension n and p        respectively,    -   A and B represent the design matrices derived from the        linearized observation equations of the GNSS model used, and    -   e is the noise vector (the residuals).

The entries of the parameter vector a are unknown integer carrier phaseambiguities, which are expressed in units of cycles, i.e. aεZ^(n). Theremaining parameters are the so-called baseline parameters, i.e.bεR^(p)including for instance atmospheric delays (see [1], sections 2.2and 3.1). When the number of observations increases, the noise vectorbecomes to behave in accordance with a normal distribution.

The observation vector y may comprise both phase and code observations,on as many frequencies as available, and accumulated over allobservation epochs. The problem described in above section “[2. Basicprinciples of the GNSS integer resolution problem]” is thereforegeneralized with many observations, based on one or more frequenciesfrom a plurality of satellites for determining the position of areceiver.

The observation equations may take into account many observation dataand many types of observation data. For instance, differential dataobtained from a differential GPS (DGPS) system. DGPS uses one or morereference stations whose positions are precisely known. This enables tocompute the effect of the ionosphere, satellite clock drifts, satelliteephemeris error at one moment in time. DGPS techniques were originallydeveloped to compensate for the intentional clock error added when theso-called “Selective Availability (SA)” feature of the GPS was turnedon. The SA feature has been permanently turned off in 2000, but DGPStechniques are still useful to compensate for effects which cause areasonably constant delay during a certain period of time. Theinformation can therefore be provided to the receiver, which can take itinto account for improving the position estimation precision. In otherwords, the DGPS reference stations transmit differential corrections tobe used by the GNSS receiver to improve the position estimation. Thesecorrections may be integrated as observations into the observationequations without affecting the general principles behind the GNSSinteger ambiguity resolution problem.

The system of equations (1) and the problem associated with itsresolution are particular for the following reason. The equationsinvolve real number unknowns (baseline parameters b) and integer numberunknowns (integer ambiguities a). Techniques are not well-established tohandle the resolution of a system of equations where some of theunknowns are known to have integer values.

In the above observation system (1), the number of unknowns is not equalto the number of observations. There are more observations than thenumber of unknowns, so that the system is an overdetermined system ofequations. Statistical properties can be associated to each possiblesolution of the overdetermined system. The resolution of the systemconsists in finding the most likely solution to the system, includingvalues for the integer number unknowns and real number unknowns.

One specific characteristic of carrier phase measurement based GNSSmodels is that, if an integer solution can be identified with a highconfidence (i.e. with a probability of correct solution close to one),this provides a very precise solution (up to centimeter-levelprecision). However, integer ambiguity resolution carries the risk thatthe carrier phase ambiguities on one or more satellites and frequencybands are incorrectly fixed. Because GNSS GPS signals have a carrier ofabout 20 centimeters, if a carrier phase ambiguity is set to a wronginteger, this may result in position errors of decimeters or more.

3.2 Generalization in Time, and Float Solution

In above section “[2. Basic principles of the GNSS integer resolutionproblem]”, only one snapshot in time was considered. However, theprecision may be improved over time, by using a series of successiveobservations.

The phase of the carrier broadcasted from one satellite differs from oneobservation at a first point in time to another observation at a secondpoint in time. However the carrier phases may be tracked, so that thecarrier phase ambiguities themselves do not change. In other words, thereceiver can lock onto the phase of one carrier.

A usual technique to solve the system of equations (1) is to first treatthe unknowns a as real numbers, i.e. floating-point numbers (even thoughthe unknowns of the vector a are known to be integer numbers). Thisapproach is advantageous because well-known techniques such as Kalmanfiltering and least squares resolution may be used to derive a floatsolution, i.e. a set of real numbers for the unknown carrier phaseambiguities. This approach is advantageous even though it does not takeinto account the fact that the unknown carrier phase ambiguities, i.e.the entries of the parameter vector a in the equation system (1) above,are integer values.

The approach is also advantageous because it converges to the correctinteger solution, provided that sufficient time is allowed. By addingsnapshots (sets of observations at respective epochs), and by keepingobservations associated with each snapshot to improve the solution (e.g.least-squares solution) a converging solution emerges.

However, a practical disadvantage of the float solution is that, whileit converges, it takes time to do so. Typically, float solutionsgenerally takes tens of minutes to converge sufficiently forcentimeter-level work. While the Kalman filter may be used to obtain afloat solution, and the float solution converges to a good estimate ofthe carrier phase ambiguities, it takes a long time to do so. There is aneed to speed up this process.

The fact that the unknowns a are integer numbers may be used asconstraints, for deriving combinations of integer values from the floatsolution, even before the float solution has converged sufficiently tothe correct integer solution. This will now be explained.

3.3 Fixing the Float Solution to Integer Values: Fixed Solution

Known approaches exist to fix the ambiguities to integer values. Forinstance, the float solution is projected or mapped to an integersolution. See sections 2.1 and 2.2 of Teunissen, P. J. G. (2003), GNSSBest Integer Equivariant estimation, presented at IUGG2003, session G04,Sapporo, Japan (herein referred as “[3]”).

This enables to obtain a fixed solution quicker. The reason behindfixing the float solution to an integer solution is to use the knowledgethat the ambiguities in fact must be integer so that, if the correctintegers are selected based on the float values of the ambiguities, theestimation of the remaining unknowns (i.e. the unknowns of vector b inequation system (1)) is improved and accelerated. The ambiguities aretherefore fixed to the most probable integer solution, to reduce thenumber of unknowns and to thereby increase the over-determination of thesystem of equations.

Fixing includes setting each ambiguity to an integer value and using thecombination of integer values to reduce the number of unknowns into thesystem of equations. However, while fixing the ambiguities speeds up theconvergence process, this comes at the risk of incorrectly fixing theambiguities, which can lead to the convergence towards a wrong solution.

It is noted that the ambiguities indicate the unknown number of cyclesof the carrier between the receiver and the satellite at a particularinstance in time, e.g. when initializing the system or at any otherstarting point in time, and thus are fixed values. The filter thus isused to estimate these fixed values based on the observations made. Moreprecisely, by collecting more and more observations over time, the statevector of the filter, comprising inter alia the ambiguities, graduallyconverges to stable integer values representing the ambiguities.

To account for the changing distance from the satellite to the receiverover time usually a phase locked loop is employed in the receiver totrack the carrier signal to determine the additional number of cycles tobe added or deducted from the initial value that is to be estimated bythe filter for the ambiguities at the starting point in time.

The Best Integer Equivariant (BIE) estimator (see section 4 “BestInteger Equivariant Estimation” of [3], or chapter 4 of [1]) is anexample of estimator in which the integer-nature of the carrier phaseambiguities is used without explicitly enforcing a single correctinteger ambiguity combination. The BIE approach uses a weighted averageof integer ambiguity combinations to produce a solution which has aprecision that is always better than or as good as the precision of itsfloat and fixed counterparts (see [1], section 4.1, page 69).

An important computational aspect of the BIE approach is that the floatambiguity solution should be transformed into a more orthogonal space toaccelerate the process used to generate integer ambiguity combinationsused in the weighted summation (see [1], section 4.2.2, page 71). Thistransformation into a more orthogonal search space is referred to as“Z-transformation” and is explained in more details for instance in [1],pages 33-36; section 3.1.4 or in Teunissen, The least-squares ambiguitydecorrelation adjustment: a method for fast GPS integer ambiguityestimation, Journal of Geodesy, 70: 65-82. A two-dimensional example ofZ-transformation of an ambiguity search space is shown in [1], page 34,FIG. 3.6, in page 34. As mentioned in [1], page 34, lines 3-6: “Due tothe high correlation between the individual ambiguities, the searchspace in the case of GNSS is extremely elongated, so that the search forthe integer solution may take very long. Therefore, the search space isfirst transformed to a more spherical shape by means of a decorrelationof the original float ambiguities.”

The Z-transformation, also called herein simply Z-transform, isdifferent from (and should not be confused with) a Fourier transform inthe frequency domain. The Z-transform is also different from (and shouldnot be confused with) the mapping of the float values from a R^(n) realnumber space to a Z^(n) integer number space.

4. Weighting Over the Ambiguities

Methods have also been disclosed to estimate the receiver position basedon all possible integer combinations.

Betti B., Crespi M., Sanso F., A geometric illustration of ambiguityresolution in GPS theory and a Bayesian approach, Manuscripta Geodaetica(1993) 18:317-330 (herein referred as “[4]”) discloses a method whereinthe ambiguities do not need to be resolved, but rather the methodinvolves “sum[ming] over all possible ambiguities with proper weightsdirectly derived from the likelihood function” (page 326, left-handcolumn, Remark 4.2).

In [4], it is also suggested to restrict the averaging to some of thecombinations of integer values: “It has to be underlined that on thepractical implementation of . . . in reality we have extended thesummation not over the whole grid of ambiguities but just to the closerknots as the function . . . drops very quickly to zero when β attainslarge values” (page 327, right-hand column, lines 34-39).

5. Problem to be Solved

There is a need for improving the implementation of positioning systemsbased on GNSS carrier phase measurements, to obtain a precise estimationof the receiver position in quick, stable and user-friendly manner.

SUMMARY

Embodiments of the present invention aim at meeting the above-mentionedneeds. In particular, embodiments of the invention aim at improving theimplementation of the methods of the prior art with in mind the goals ofobtaining rapidly a stable and precise solution, while improvingusability.

Embodiments of the invention include methods, apparatuses, rovers,network stations, computer programs and computer-readable mediums, asdefined in the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention shall now be described, inconjunction with the appended Figures in which:

FIG. 1 schematically shows a GNSS with two satellites and one receiver,for illustrating the background of the invention

FIG. 2 a schematically shows an uncertainty region wherein the positionof the receiver is known to be located, for illustrating the backgroundof the invention;

FIG. 2 b schematically shows the uncertainty region of FIG. 2 a whereinwave fronts associated with a first satellite, Satellite 1, arerepresented for illustrating, within the background of the invention, acarrier phase ambiguity;

FIG. 2 c schematically shows the uncertainty region of FIG. 2 a and thewave fronts of FIG. 2 b, wherein wave fronts associated with a secondsatellite, Satellite 2, are represented for illustrating, within thebackground of the invention, two carrier phase ambiguities;

FIG. 2 d schematically shows the uncertainty region of FIG. 2 a and thewave fronts of FIGS. 2 b and 2 c, wherein wave fronts associated with athird satellite, Satellite 3, are represented for illustrating, withinthe background of the invention, three carrier phase ambiguities;

FIG. 2 e is identical to FIG. 2 d, except that a search node,corresponding to a combination of integer carrier phase values isschematically illustrated;

FIG. 3 schematically shows an uncertainty region wherein the position ofthe receiver is known to be located, the wave fronts associated with twosatellites and the probability associated with each integercombinations, i.e. each set of integer values for each ambiguity, forillustrating the background of the invention;

FIG. 4 is identical to FIG. 3 except that two search nodes are shown tohave substantially the same probability of corresponding to the correctsolution, for illustrating the background of the invention;

FIG. 5 a is a flow chart illustrating one embodiment of a method of theinvention;

FIG. 5 b illustrates exemplary benefits in terms of convergence speed ofone embodiment of a method of the invention;

FIGS. 6 a and 6 b are two flowcharts, each illustrating the steps ofassigning integer values to the float solution to form candidate setsand selecting candidate sets based on the first threshold, within oneembodiment of a method of the invention;

FIG. 7 illustrates the process of searching a tree of integerambiguities to select candidate sets within one embodiment of a methodof the invention;

FIG. 8 is a flow chart illustrating a portion of one embodiment of amethod of the invention;

FIG. 9 is a flow chart illustrating a portion of one embodiment of amethod of the invention, wherein, if the number of originally selectedcandidate sets is not sufficient, more candidate sets are selected;

FIG. 10 is a flow chart illustrating a portion of one embodiment of amethod of the invention, wherein, if the number of originally selectedcandidate sets is larger than necessary, some candidate sets areexcluded;

FIGS. 11 a, 11 b and 11 c are three flow charts illustrating threeembodiments of a method of the invention, wherein adapted qualitymeasures are used to influence the formation of the weighted average;

FIG. 12 illustrates a portion of the flowcharts of FIGS. 11 a to 11 c;

FIGS. 13 a and 13 b illustrate details of one of the steps, i.e. step1170, as illustrated in FIGS. 11 a to 11 c;

FIG. 14 illustrates the scaling of the probability distribution ofresiduals before selecting and/or forming the weighted average of thecandidate sets;

FIG. 15 is a flowchart illustrating an embodiment of a method of theinvention, wherein a formal precision and an achieved precision aredetermined to provide an indication of the convergence of the positionestimation process;

FIG. 16 a illustrates a portion of the flowchart of FIG. 15 in aparticular embodiment of the method illustrated in FIG. 15;

FIGS. 16 b to 16 e illustrate examples of embodiments for providing anindication of convergence (e.g. “float” or “fixed”) depending on theconvergence of the achieved precision with respect to the formalprecision;

FIGS. 17 a and 17 b illustrate a flowchart portion of a method of oneembodiment of the invention, wherein unconditional inclusion andexclusion thresholds respectively are used in the process of providingan indication of the convergence of the iFlex solution;

FIGS. 18 a to 18 d illustrate examples wherein an indication ofconvergence is provided based on the achieved precision in embodimentsderived from those described with reference to FIGS. 17 a and 17 b;

FIGS. 19 and 20 are two flowcharts illustrating two embodiments of amethod of the invention, wherein satellite legacy observations are keptafter interruption of the tracking of the satellite signals in view ofimproving the position estimation;

FIGS. 21 and 22 are two flowcharts illustrating two embodiments of amethod of the invention, wherein a subset of the ambiguities, and notall observed ambiguities are taken into account in the filter statevector for forming the float solution;

FIG. 23 is a block diagram of an integrated GNSS receiver system inaccordance with some embodiments of the invention;

FIG. 24 schematically illustrates a network positioning scenario inaccordance with some embodiments of the invention; and

FIG. 25 schematically illustrates a real-time-kinematic positioningscenario in accordance with some embodiments of the invention;

DETAILED DESCRIPTION

The present invention shall now be described in conjunction withspecific embodiments. It may be noted that the specific embodimentsserve to provide the skilled person with a better understanding, but arenot intended to in any way restrict the scope of the invention, which isdefined by appended claims. In particular, the embodiment describedindependently throughout the description can be combined to form furtherembodiments to the extent that they are not mutually exclusive.

1. Candidate Set Selection

It has been notably recognized by the inventors that efficientlyselecting a number of candidates for the weighted average may improvethe estimation at reduced processing requirements.

According to one embodiment, the integer solutions, i.e. integercandidate sets, used to form the weighted average are selected on thebasis of a quality measure. More precisely, candidate sets can beselected that have a quality measure better than a threshold determinedbased on a reference quality measure of a reference candidate set.

Thus, the quality measure, e.g. statistical distance of the candidateset having the best quality measure relative to the “float solution” ofthe ambiguities in the state vector of the filter, determines theselection. Thus, the threshold used for the selection of candidate setsto be used in the weighted average may depend on the quality measure ofthe best candidate set, in other words, depends on how far,statistically, the best candidate is from the float solution. Thefurther the best candidate set is from the float solution, the lower thethreshold can be set so that a larger number of candidate sets isselected for the weighted average as compared to a best candidate setthat is, statistically, close to the float solution. This improves thereliability of the weighted average.

The advantages of this procedure include:

-   -   there is no need to fix the integer ambiguities at one point, so        that the risk of wrongly fixing the integer ambiguities is        eliminated;    -   the weighted average of the selected candidate sets converges        more rapidly to the correct integer solution, so that there is        no need to wait tens of minutes before being able to carry out        centimeter-level work; and    -   the precisions provided by the weighted average of the selected        candidate sets are more realistic than those reported by fixing        solutions consisting in fixing at one point in time some of the        carrier phase ambiguities (in equation (1)) in attempt to speed        up the float solution convergence.

According to one embodiment, estimating parameters, i.e. ambiguities,derived from the GNSS signals, includes obtaining observations of a GNSSsignal from each of a plurality of GNSS satellites; feeding theobservations to a filter having a state vector at least comprising afloat ambiguity for each received frequency of the GNSS signals, eachfloat ambiguity constituting a real number estimate associated with aninteger number of wavelengths of the GNSS signal between a receiver ofthe GNSS signal and the GNSS satellite from which it is received, andthe filter being for estimating a float value for each float ambiguityof the state vector; assigning integer values to at least a subgroup ofthe estimated float values to define a plurality of integer ambiguitycandidate sets; selecting a first number of candidate sets having aquality measure better than a first threshold, wherein the firstthreshold is determined based on a reference quality measure of areference candidate set; and obtaining a weighted average of theselected candidate sets, each candidate set weighted in the weightedaverage based on its quality measure.

The ambiguities in subsequent operations can be used to determine aposition of the receiver or can be used to prepare data at a networkside (e.g., in a network processor) that can be used to augment positioninformation of a rover.

FIG. 5 a is a flow chart illustrating one embodiment of a method of theinvention. The method 100 illustrated by the flowchart of FIG. 5 a is amethod for estimating parameters derived from GNSS signals useful todetermine a position.

The parameters to be estimated are the unknown parameters of the unknownparameter vector described in relation to equation (1) (see backgroundsection). That is, the parameters include the unknown integer carrierphase ambiguities, forming the vector a, and the real number baselineparameters, forming the vector b.

There is one unknown integer carrier phase ambiguity, i.e. one unknowninteger number of cycles, per observed carrier (i.e. per observedfrequency, or more generally per frequency for which observations areobtained) per satellite. If each satellite transmits three carriers,i.e. for instance a 1575.42 MHz carrier (GPS L1 frequency band), a1227.60 MHz carrier (GPS L2 frequency band), and a 1176.45 MHz carrier(GPS L5 frequency band), and if all three frequencies are observed (ormore generally if observations are obtained for all three frequencies),if the signals from 10 GNSS satellites are observed (or more generallyif observations are obtained for 10 GNSS satellites), there are 30unknown integer carrier phase ambiguities. In other words, the dimensionof the vector a is 30.

It is possible to observe (or more generally to obtain observations for)only one frequency per satellite, or to observe (or more generally toobtain observations for) two or more frequencies per satellite,depending on the receiver and the satellites. However, it is beneficialto observe (or more generally to obtain observations) and use as manyfrequencies as available, since this increases the over-determinedcharacter of the system.

There are as many baseline parameters as provided by the GNSS modelwhich is used. The dimension of the vector b depends on the GNSS modelused. A typical GNSS model includes but is not limited to parameters forposition (x, y, z), time (t), and atmospheric influences such asionosphere and troposphere.

The GNSS signals include signals from GNSS satellites. This may includethe simultaneous use of signals from satellites belonging to differentGNSS infrastructure, i.e. for instance signals from both GPS and GLONASSsatellites.

The method, the parameters to be estimated and the GNSS signals areuseful for determining the position of a GNSS receiver.

The method includes a step of, or procedure for, obtaining observations(step 120) of the GNSS signals from the satellites. This includesreceiving, i.e. acquiring, the signals and determining, or measuring, atone point in time the carrier phase of the GNSS signals coming from theGNSS satellites. This also includes tracking the carrier phase in time.This may also include determining the time of arrival of a code comingfrom a GNSS satellite on one frequency. This may also include retrievingobservations which have been previously measured and stored for laterprocessing (post processing). The step of, or procedure for, obtaining120 observations generates or retrieves, i.e. outputs, a series ofobservations constituting the vector y of equation (1). The observationsare obtained not only at one point in time, but are measuredsequentially (though they may be retrieved from storage for batch ratherthan sequential processing).

In step or procedure 140, the observations are fed into a filter, suchas a Kalman filter, which uses the observations to estimate the value ofunknown parameters. The filter has a state vector containing a value foreach one of the unknown parameters. The observations are fed into thefilter and the state vector is updated accordingly. For near real-time(sometimes called real-time) processing the observations are typicallyprocessed sequentially by epoch, while for post processing theobservations of multiple epochs may be processed together.

The state vector at least includes a real number estimate for each oneof the unknown integer carrier phase ambiguities. Each of these realnumber estimates of integer carrier phase ambiguities is called herein a“float ambiguity”.

There is one float ambiguity per satellite for each received frequencyof the GNSS signals. The float ambiguity for a received frequency of aGNSS signal is a real number estimate associated with an integer numberof wavelengths, or cycles, of the GNSS signal between a receiver of theGNSS signal and the GNSS satellite from which it is broadcasted. A floatambiguity may be a real number estimate of the integer number ofwavelengths or cycles of the GNSS signal between the receiver and theGNSS satellite.

A float ambiguity may also be a real number estimate of the integerdifference between a first integer number of cycles of the GNSS signalbetween the receiver and a first GNSS satellite and a second integernumber of cycles of the GNSS signal between the receiver and a secondGNSS satellite (single-differenced ambiguity). A float ambiguity mayalso be a real number estimate of the integer difference between a firstinteger number of cycles of the GNSS signal between a first receiver anda GNSS satellite and a second integer number of cycles of the GNSSsignal between a second receiver and the same GNSS satellite(single-differenced ambiguity).

A float ambiguity may also be a real number estimate of the integerdifference of differences (double-differenced ambiguity); that is afirst single-differenced ambiguity is formed e.g. as the differencebetween a first integer number of cycles of the GNSS signal between afirst receiver and a first GNSS satellite and a second integer number ofcycles of the GNSS signal between a second receiver and the first GNSSsatellite; a second single-differenced ambiguity is formed e.g. as thedifference between a third integer number of cycles of the GNSS signalbetween the first receiver and a second GNSS satellite and a fourthinteger number of cycles of the GNSS signal between the second receiverand the second satellite; and a double-differenced ambiguity is formedas the difference between the first single-differenced ambiguity and thesecond single-differenced ambiguity.)

Whether undifferenced, single-differenced, or double-differenced, thisis why the float ambiguity for a received frequency is defined as a realnumber estimate associated with an integer number of wavelengths orcycles.

The filter estimates a float value for each float ambiguity of the statevector. The output of the filter is a float solution, which includes afloat value for each float ambiguity of the state vector.

The filtering process can be readily implemented in terms of a Kalmanfilter, sequential least squares estimator, robust estimation, or othercomparable data processing schemes. Kalman filtering techniques are wellestablished in the field of GNSS data processing and are able togenerally handle state descriptions where one or more parameter hastime-variant properties. A description of Kalman filtering applied toGPS data processing can be found in “Introduction to Random Signals andApplied Kalman Filtering”, Brown, R. G, & Hwang, P. Y. C., John Wiley &Sons, 3^(rd) Ed, ISBN: 0-471-12839-2.

The float solution is obtained by allowing the integer carrier phaseambiguities to take real values. The constraint that the carrier phaseambiguities are integer numbers is then applied. That is, an integervalue is assigned to each of the float values forming the floatsolution. In step or procedure 160, integer values are assigned to eachof the estimated float values to define a plurality of integer ambiguitycandidate sets. Since there is more than one way to assign integervalues to the float values of the float solution, step 160 leads todefining a plurality of integer ambiguity candidate sets, i.e. aplurality of combination of integer values corresponding to a certainextent to the float solution.

For instance, let us imagine that the float solution comprises sevenfloat values, because seven frequencies are observed (or, moregenerally, observations are obtained for seven frequencies). The floatsolution at one point in time may for instance be constituted by

-   -   (2.11, 3.58, −0.52, −2.35, 1.01, 0.98, 1.50)

In this example, the float value “2.11” is a real-number estimateddifference between

-   -   the number of cycles from a first GNSS satellite on a first        frequency to the receiver, estimated as a real number by the        filter of step 140, and    -   the number of cycles from the first satellite on the first        frequency to the receiver, as estimated by a rough estimation        method (for instance determined by code-based GNSS positioning).

The so-called rough estimation method may for instance provide theboundary of an uncertainty region as illustrated in FIGS. 2 a to 2 e,wherein the centre of the uncertainty region (which may be a circle) isconsidered to be the number of cycles as estimated by the roughestimation method. If the estimated difference is the float value“2.11”, it means that the float solution includes for the firstambiguity a number of cycles differing by 2.11 cycles from the centre ofthe uncertainty region.

The float value “2.11” is therefore a real number estimate associatedwith the integer number of cycles from the first satellite on the firstfrequency to the receiver.

There are many possibilities to form sets of integers from the floatsolution. In other words, projecting the float solution from R^(n), an-dimensional space with real coordinates, into Z^(n), a n-dimensionalspace with integer coordinates, may result in many different integervectors, or integer candidate sets.

One simple way to project the float solution into the integer space isto round each ambiguity, i.e. each float value of the float solution, toits closest integer value. This gives a first ambiguity candidate set:

-   -   (2, 4, −1, −2, 1, 1, 1)        if the last float value “1.50” is rounded down to “1”. If the        float value “1.50” is instead rounded up to “2”, a second        ambiguity candidate set is provided:    -   (2, 4, −1, −2, 1, 1, 2)

The second float value “3.58” and the third float value “−0.52” may alsobe rounded to “3” and “0” respectively (with still in mind the goal ofminimizing the residuals e of equation system (1)). This gives anadditional series of six candidate sets:

-   -   (2, 3, −1, −2, 1, 1, 1)    -   (2, 4, 0, −2, 1, 1, 1)    -   (2, 3, 0, −2, 1, 1, 1)    -   (2, 3, −1, −2, 1, 1, 2)    -   (2, 4, 0, −2, 1, 1, 2)    -   (2, 3, −0, −2, 1, 1, 2)

The optional step 140 a of performing a Z-transformation, as illustratedin FIG. 5 a, will be discussed below.

Step 160 consists in forming, or defining, these candidate sets ofinteger values, by assigning integer values to the float values. Thecandidate sets of integers are called integer ambiguity candidate sets.

The step 160 of forming the integer ambiguity candidate sets includesdefining a plurality of candidate sets. How many candidate sets todefine may be determined based on one of several criteria.

In one embodiment, all candidate sets falling within an uncertaintyregion of the type shown in FIGS. 2 a to 2 e are formed by the assigningstep 160. This embodiment has the advantage that, if it is known with avery high probability that the correct combination of integers fallswithin the uncertainty region, it is also known with an equally highprobability that one amongst the considered candidate sets is thecorrect one. The number of candidate sets may however become large. Forinstance, if the uncertainty region includes six possible integer valuesfor each ambiguity (a situation which corresponds approximately to theone shown in FIGS. 2 a to 2 e), and if there are seven ambiguities ineach integer ambiguity candidate set (FIGS. 2 d and 2 e only show threeambiguities), there are 6⁷ (six raised to the power of seven) candidatesets, which gives 279936 candidate sets.

Other embodiments of the method and step 160 in particular will bedescribed with reference to FIGS. 6 a and 6 b.

The above-described example involves seven ambiguities in each integerambiguity candidate set. This is however only an example and there maybe more than or fewer than seven ambiguities and corresponding floatvalues.

Still referring to FIG. 5 a, the step 180 of selecting candidate setsincludes

-   -   identifying one reference candidate set amongst the candidate        sets defined in step 160,    -   calculating a reference quality measure of the reference        candidate set, wherein the reference quality measure represents        how close the candidate set is from the float solution,    -   defining an inclusion threshold, herein referred to as first        threshold, based on the reference quality measure, and    -   selecting, amongst the candidate sets defined in step 160, those        which have a quality measure better than the first threshold.

The selected candidate sets constitute the output of step 180. The firstthreshold is determined based on how close the reference candidate setis from the float solution. This enables the process to be tailored tohow good the best candidate set is, which has been recognized as being ameasure of the degree of convergence of the estimation process.

Step 200 includes forming a weighted average of all the candidate setsselected in step 180, wherein the weight associated with each candidateset is determined based on the quality measure, or probability ofcorrectness, of the candidate set. The likelier a candidate set is to becorrect, the larger is its weight in the weighted average. This weightedaverage forms a new float solution, called herein the iFlex solution.The iFlex solution is a weighted combination of some of the possibleinteger ambiguity outcomes, not all of them (within the uncertaintyregion). The iFlex solution has been found to converge more rapidly tothe correct solution in a reasonable amount of computation time ascompared with the convergence time of the float solution.

The optional step 200 a of performing a reverse Z-transformation, asillustrated in FIG. 5 a, will be discussed below.

The quality measure reflects how well one particular integer ambiguitycandidate fits the observations. Hence the quality measure isproportional to the size of the observation residuals.

A suitable quality measure Ψ_(â) _(k) can be computed via the followingmatrix inner product:Ψ_(â) _(k) =(â−â _(k))^(T) Q _(â) ⁻¹(â−â _(k))  (2)where â is the vector of float ambiguity values; â_(k) is thecorresponding vector of integer ambiguity values for candidate set k;Q_(â) ⁻¹ is the inverse of the float ambiguity covariance matrix. Thescalar expression above can be referred to as a residual error norm.

For instance, if the above-mentioned exemplary eight candidate sets areselected during step 180, and the quality measure is calculated for eachof them based on the above-mentioned float solution (2.11, 3.58, −0.52,−2.35, 1.01, 0.98, 1.50), a weight is then computed for each candidateset based on the quality measure, and the weighted average is calculatedbased on the candidate sets and weights. Table 1 represents an exampleof selected candidate sets and associated quality measures.

The quality measure values illustrated in Table 1 are computed accordingto the expression for Ψ_(â) _(k) above. Assuming that the measurementsare normally distributed, the corresponding candidate weights ω(â_(k))can be computed according to the following expression:

$\begin{matrix}{{\omega\left( {\overset{\Cap}{\alpha}}_{k} \right)} = \frac{\exp\left\{ {{- \frac{1}{2}}\Psi_{{\overset{\Cap}{\alpha}}_{k}}} \right\}}{\sum\limits_{{\overset{\Cap}{\alpha}}_{k} \in Z^{n}}{\exp\left\{ {{- \frac{1}{2}}\Psi_{{\overset{\Cap}{\alpha}}_{k}}} \right\}}}} & (3)\end{matrix}$

TABLE 1 (example). Candidate Set index Quality Weight (k) Selectedcandidate sets measure ω({circumflex over (α)}_(k)) 1 (2, 4, −1, −2, 1,1, 1) 0.219 0.9678 2 (2, 4, −1, −2, 1, 1, 2) 8.719 0.0138 3 (2, 3, −1,−2, 1, 1, 1) 9.819 0.0079 4 (2, 4, 0, −2, 1, 1, 1) 11.102 0.0042 5 (2,3, 0, −2, 1, 1, 1) 11.559 0.0033 6 (2, 3, −1, −2, 1, 1, 2) 13.299 0.00147 (2, 4, 0, −2, 1, 1, 2) 13.859 0.0011 8 (2, 3, 0, −2, 1, 1, 2) 15.5980.0004

The weighted average of the integer ambiguity candidates ({right arrowover (a)}_(iFlex)) is computed via the following expression:

$\begin{matrix}{{\overset{\rightharpoonup}{a}}_{iFlex} = {\hat{a} + {\sum\limits_{{\overset{\Cap}{\alpha}}_{k} \in Z^{n}}{\left( {\hat{a} - {\overset{\Cap}{\alpha}}_{k}} \right){\omega\left( {\overset{\Cap}{\alpha}}_{k} \right)}}}}} & (4)\end{matrix}$

Based on the values given in Table 1, the iFlex ambiguity estimates areas follows:

(2.0000, 3.9869, −0.9910, −2.0000, 1.0000, 1.0000, 1.0167)

A suitable quality measure to be used to form the weights in theweighted average has been described above. The quality measure mayhowever be based on various weighting functions. Notably, a normal(distribution of the type

$\left. {\mathbb{e}}^{{- \frac{1}{2}}{\sum{(\frac{x - \mu}{\sigma})}^{2}}} \right),$a Laplace (distribution of the type

$\left. {\mathbb{e}}^{{- \alpha}{\sum{\frac{x - \mu}{\sigma}}}} \right)$or a minmax (distribution of the type e^(−max|x−μ|)) probabilitydistribution may be used to compute the probability of correctness ofthe candidate sets and thus the weights.

In one embodiment (notably illustrated by the optional step 140 a inFIG. 5 a), after step 140 but before step 160, the float solution istransformed to a more orthogonal space, for instance using a Z-transformas disclosed at the end of section 3.3 of the background section. From acomputational standpoint, the Z-transform approach enables toefficiently cover the integer search space. Although the iFlex techniquemay be carried out without the Z-transform, it is advantageous to usethe Z-transform for a practical implementation. Other decorrelationtechniques may be used. If a Z-transformation is carried out on thefloat solution, a reverse Z-transformation should also be performed, asillustrated by the optional step 200 a in FIG. 5 a. Although notillustrated in all Figures, such as on FIG. 8, the Z-transformation andthe reverse Z-transformation also constitute optional and advantageoussteps in the context of the other embodiments.

FIG. 5 b illustrates the benefits of one embodiment of the method of theinvention in terms of convergence speed and precision. The graph showsthe convergence in terms of position error obtained as a function oftime. Three methods may be compared by referring to the graph.

Firstly, the float solution is shown to converge to the correct integersolution, but the convergence is slow (as shown on FIG. 5 b by the curvereferred to by the label “Float Solution”). Secondly, a method consistsin fixing some of the carrier phase ambiguities of the float solution tointeger values to rapidly constrain the float solution (i.e. by fixingsome of the unknowns of vector a to increase the over-determinedcharacter of the system (1)). This second method may rapidly lead to acorrect fixed solution with a good precision (as shown on FIG. 5 b bythe curve referred to by the label “Correct Fixed Solution”). However,the second method may lead to an incorrect fixed solution showing alarge position error (as shown on FIG. 5 b by the curve referred to bythe label “Incorrect Fixed Solution”). Finally, the iFlex solution basedon the method described with reference to FIG. 5 a fills the gap betweenthe float solution (converging but slowly) and the fixing solution(converging rapidly, but with risks of errors), by providing anintermediate solution, with rapid convergence and low risk of errors.

In the method described with reference to FIG. 5 a, steps 160 and 180need not be performed successively. That is, some sub-steps of step 160may be performed after some sub-steps of step 180. In other words, theremay be an overlap when performing steps 160 and 180. This will beexplained further with reference to FIGS. 6 a and 6 b.

FIG. 6 a illustrates one embodiment of steps 160 and 180 for assigninginteger values to the float solution and selecting candidate sets as aninput for the weighted average formed in step 200. The steps 120, 140and 200 of FIG. 5 a are not illustrated in FIG. 6 a for the sake ofclarity.

In step 160 a, one reference candidate set is formed by assigninginteger values to the float solution. This may for instance be performedin a straightforward manner by rounding each float value of the floatsolution. This reference candidate set is included within the group ofselected candidate sets (step 180 a). At this stage, there is only onecandidate set, namely the reference candidate set, in the group ofselected candidate sets. In step 180 b, the reference quality measure,which is the quality measure of the reference candidate set defined instep 160 a, is calculated. Based on the reference quality measure, aninclusion threshold, or first threshold, is determined (step 180 c). Theinclusion threshold will serve to decide whether to include each one ofthe other candidate sets within the group of selected candidate sets.

An additional candidate set is then formed by assigning integer valuesto the float solution (step 160 b). The additional candidate set definedin step 160 b differs from the reference candidate set defined in step160 a. The differences between the additional candidate set and thereference candidate set relate to at least one ambiguity. The qualitymeasure of the additional candidate set is calculated (step 180 d) andthis quality measure is compared (step 180 e) to the inclusionthreshold. If the quality measure of the additional candidate set isbetter than the inclusion threshold, the additional candidate set isincluded (step 1800 within the group of selected candidate sets.Otherwise, the additional candidate set is not included in the group ofselected candidate sets (step 180 g).

The steps 160 b, 180 d, 180 e and 180 f/180 g are repeated until thereare no more candidate sets to examine, or until a maximum number ofselected candidate sets has been reached. The maximum number of selectedcandidate sets to select may be predetermined, i.e. determined inadvance.

FIG. 6 b illustrates another embodiment of steps 160 and 180 forassigning integer values to the float solution and for selectingcandidate sets as an input for forming the weighted average in step 200.The steps 120, 140 and 200 of FIG. 5 a are not illustrated in FIG. 6 bfor the sake of clarity.

In step 160 c, a plurality of candidate sets are formed by assigninginteger values to the float solution. In this step, a large number ofcandidate sets may be defined. The candidate sets to be selected will beselected from this plurality of candidate sets.

Step 160 c may include rounding each float value of the float solutionto form a first candidate set, and by forming the other candidate setsby modifying the integer values assigned to each ambiguity within acertain range (i.e. for instance within an uncertainty region, asillustrated in FIGS. 2 a to 2 e). In other words, step 160 c may providea large number of candidate sets determined by combination ofmodifications (adding . . . , −2, −1, 0, 1, 2, . . . ) of the integervalues of first candidate set.

One candidate set formed in step 160 c is selected (step 180 h) toconstitute one reference candidate set. The selection of the referencecandidate set may be carried out in the manner described with referenceto steps 160 a of FIG. 6 a.

The reference candidate set selected in step 180 h is included (step 180a) within the group of selected candidate sets. In step 180 b, thereference quality measure, which is the quality measure of the referencecandidate set, is calculated. Based on the reference quality measure, aninclusion threshold, or first threshold, is determined (step 180 c). Theinclusion threshold will serve to decide whether to include each one ofthe other candidate sets (the candidate sets formed in step 160 c, butnot selected as reference candidate set in step 180 h) within the groupof selected candidate sets.

A candidate set, which belongs to the candidate sets formed in step 160c and which has not been included yet in the group of selected candidatesets, is considered (step 180 i).

The quality measure of the considered candidate set is calculated (step180 d) and this quality measure is compared (step 180 e) to theinclusion threshold. If the quality measure of the considered candidateset is better than the inclusion threshold, the considered candidate setis included (step 1800 within the group of selected candidate sets, toform a new selected candidate set. Otherwise, the considered candidateset is not included in the group of selected candidate sets (step 180g).

The steps 180 i, 180 d, 180 e and 180 f/180 g are repeated until thereare no more candidate sets to consider, or until a maximum number ofselected candidate sets has been reached, as explained in a similar waywith reference to FIG. 6 a.

FIG. 7 illustrates another exemplary implementation of the steps 160 and180 of assigning integer values to the float solution to form integercandidate sets and selecting candidate sets to be used for the weightedaverage formed in step 200.

Referring to FIG. 7, the integer values and their modifications areorganized in a tree. Each level in the tree corresponds to oneambiguity. The tree can be walked through to select the candidate setsthat will form the basis of the weighted average. In one embodiment, thereference candidate set is formed by rounding each individual floatvalue, which corresponds to taking all left-hand side branches of thetree. In the illustrated example, this gives the candidate sets

-   -   (2, 4, −1, . . . , 1)

Then, for the selecting step 180, all integer modifications areconsidered within an uncertainty region, first

-   -   (2, 4, −1, . . . , 2)        then:    -   (2, 4, −1, . . . , 0)

In the same manner, all branches which have a good expectation of beingselected are analyzed. The quality measure for each of the nodes iscalculated according to:Ψ_(â) _(k) =(â−â _(k))^(T) L _(ã) ^(T) D _(ã) L _(ã)(â−â _(k))  (5)

In which the matrix product L_(ã) D_(ã)L_(ã) ^(T) is derived from thefactorization of the inverse float ambiguity covariance matrix intolower unit triangular component L_(ã) and diagonal component D_(ã) via:L_(â)D_(â)L_(â) ^(T)=Q_(â) ⁻¹  (6)The quality measure can therefore be presented in the following form:Ψ_(â) _(k) =[(â−â _(k))^(T) L _(ã) ]D _(ã) [L _(ã) ^(T)(â−â _(k))]  (7)

Because of the triangular nature of L_(ã), the quality measure valuesalways increase as lower branches of the search tree are encountered.Therefore, if the maximum acceptable quality measure is say 1.56, anybranches that start with a quality measure greater than 1.56 can beimmediately pruned.

FIG. 8 illustrates a flow chart of a further embodiment of the method,wherein the iFlex solution (formed in step 200) is used as a basis toestimate 220 the position of the receiver. The position of the receivercan be obtained by transforming the individual real number values of theiFlex solution into the position domain via the following matrixexpression:b _(IFLEX) ={circumflex over (b)}−Q _({circumflex over (b)}â) Q _(â)⁻¹(â−ā _(IFLEX))  (8)

Where Q_({circumflex over (b)}â) is the position/ambiguity correlationmatrix, {circumflex over (b)} the float position solution, â the vectorof float ambiguity estimates, āIFLEX the iFlex ambiguity estimates, andb _(IFLEX) the iFlex position estimates.

Rather than compute the iFlex solution initially in the ambiguitydomain, it is possible to form the iFlex solution directly in theposition domain by forming the following weighted average:

$\begin{matrix}{{\overset{\_}{b}}_{IFLEX} = {\hat{b} + {\sum\limits_{{\overset{\Cap}{a}}_{k} \in Z^{n}}{{\overset{\Cap}{b}}_{k}{\omega\left( {\overset{\Cap}{a}}_{k} \right)}}}}} & (9)\end{matrix}$

The weights used in (9) are the same as those used in (4) above. Theterm {circumflex over (b)}_(k) contains the three-dimensional positionin space corresponding to integer ambiguity candidate set â_(k), where:{circumflex over (b)}_(k) =−Q _({circumflex over (b)}â) Q _(â)⁻¹({circumflex over (a)}−â)  (10)

In one embodiment, the reference candidate set is the candidate sethaving the best quality measure, i.e. the one which minimizes theresidual error norm value according to:Ψ_(â) _(BEST) =min_(â) _(k) _(εZ) _(n) {(â−â_(k))^(T) Q _(ã) ⁻¹(â−â_(k))}  (11)

In one embodiment, the first threshold, i.e. the inclusion threshold(determined in step 180 c of FIG. 6 a-6 b), is determined as at leastone of

-   -   a fraction of the reference quality measure, such as 1000 times        worse the reference quality measure,    -   a multiple of the reference quality measure, and    -   a distance to the reference quality measure.

The inclusion threshold should be understood in the following context(for instance with additional reference to the above exemplary Table 1,and especially its last two columns). On the one hand, the goal of theinclusion threshold is to define a group of best candidate sets to beincluded into the weighted average. On the other hand, the larger theweights ω(â_(k)) associated with a candidate set, the better thecandidate set; and the smaller the exemplary chi square values(â−â_(k))^(T)Q_(ã) ⁻¹(â−â_(k)) (see equation (2) above) associated witha candidate set, the better the candidate set. Multiple, fraction anddistance as determination for the inclusion threshold should beunderstood in this context, i.e. the context of aiming to include thebest candidate sets into the weighted average.

In one embodiment, as illustrated in FIG. 9, if, as a result of the step180 of selecting candidate sets based on the first threshold, i.e. basedon the inclusion threshold referred to in FIGS. 6 a to 6 b for instance,the number of candidate sets is smaller than a minimum number, furtherprocessing is carried out to select more candidate sets. The minimumnumber of candidate sets is herein referred to as a second threshold.The second threshold is a threshold in terms of number of candidatesets, while the first threshold is a threshold in terms of qualitymeasure of a candidate set.

If, as a result of the selection of step 180 based on the firstthreshold, the number of selected candidate sets (referred to herein as“first number” of selected candidate sets) is determined to be smallerthan a second threshold (step 190), a step 195 of selecting furthercandidate sets is provided. In step 195, a second number of candidatesets are selected on the basis of the quality measure of the candidatesets in decreasing order starting with the non-selected candidate sethaving the best quality measure (this may be carried out by furtherwalking the tree as shown in FIG. 7). The second number is equal to thedifference between the first number of selected candidate sets (selectedin step 180) and a second threshold defining a minimum number ofcandidate sets to be included in the weighted average.

This embodiment ensures that a minimum number of sets are included inthe weighted average (formed in step 200 in FIG. 9), on which the iFlexsolution is based. It has been recognized that selecting too fewcandidate sets for forming the iFlex solution may cause it to fail toconverge. This is notably because, if too few candidate sets areselected, the iFlex solution may possibly not include a representativesample of candidate sets needed to estimate the uncertainty in thesolution.

In one embodiment, the second threshold is comprised between 5 and 30.In one embodiment, the second threshold is equal to 10.

In one embodiment, as illustrated in FIG. 10, if, as a result of thestep 180 of selecting candidate sets based on the first threshold, i.e.based on the inclusion threshold referred to in FIGS. 6 a to 6 b forinstance, the number of candidate sets is larger than a maximum number,further processing is carried out to remove some of the selectedcandidate sets from the selection obtained based on the first threshold.The maximum number of candidate sets is herein referred to as a thirdthreshold. The third threshold is a threshold in terms of number ofcandidate sets, while the first threshold is a threshold in terms ofquality measure of a candidate set.

If, as a result of the selection of step 180 based on the firstthreshold, the number of selected candidate sets (referred to herein asfirst number of selected candidate sets) is determined to be larger thana third threshold (step 191), a further step 196 of selecting isprovided. In step 196, a third number of selected candidate sets areexcluded in decreasing order starting with the selected candidate sethaving the lowest quality measure, wherein the third number isconstituted by the difference between the first number of selectedcandidate sets and a third threshold defining a maximum number ofcandidate sets to be included in the weighted average.

It has been recognized that selecting too many candidate sets forforming the weighted average and the iFlex solution does notsubstantially improve the convergence of the iFlex solution. Selectingtoo many candidate sets also increases the computation burden tocalculate the iFlex solution.

In one embodiment, the third threshold is comprised between 100 and10000, and in one embodiment the third threshold is equal to 1000.

In one embodiment, the step 180 of selecting the candidate sets isinterrupted once the third threshold is reached.

In one embodiment, the individual values of the iFlex solution are notfed back into the observation equations to fix some of the carrier phaseambiguities (constituent elements of vector a in equation (1)) toincrease the over-determined character of the system of equations. Inother words, no fixing of any of the ambiguities is injected back intothe process of computing the float solution. At each iteration, i.e.each time a new set of observations is fed into the filter, the iFlexsolution is computed anew from the generated float solution withoutusing previous information from the iFlex solution. In contrast, thefloat solution is obtained from the filter (e.g. Kalman filter) whichuses the past state vectors as part of the computation process.

In one embodiment, the above-described method includes using theweighted average to estimate a position of the receiver of the GNSSsignals. The receiver may be a rover.

In one embodiment, in one of the above-described methods, the referencecandidate set is the candidate set having the best quality measure.

In one embodiment, in one of the above-described methods, the qualitymeasure of the candidate sets is constituted by a residual error normvalue, the residual error norm value of a candidate set being a measurefor a statistical distance of the candidate set to the state vectorhaving the float ambiguities.

In one embodiment, in one of the above-described methods, the firstthreshold is determined as at least one of a fraction of the referencequality measure, a multiple of the reference quality measure, and adistance to the reference quality measure.

In one embodiment, any one of the above-described methods includes, ifthe number of selected candidate sets (referred to herein as firstnumber of selected candidate sets) is smaller than a second threshold,selecting, for forming the weighted average, a second number of furthercandidate sets on the basis of the quality measures of the candidatesets in decreasing order starting with the non-selected candidate sethaving the best quality measure, the second number being constituted bythe difference between the first number of selected candidate sets and asecond threshold defining a minimum number of candidate sets to beincluded in the weighted average.

In one embodiment, any one of the above-described methods includes, ifthe number of selected candidate sets (referred to herein as firstnumber of selected candidate sets) is larger than a third threshold,excluding a third number of selected candidate sets from forming theweighted average in decreasing order starting with the selectedcandidate set having the worst quality measure, wherein the third numberis constituted by the difference between the first number of selectedcandidate sets and a third threshold defining a maximum number ofcandidate sets to be included in the weighted average.

In one embodiment, the selected candidate set having the worst qualitymeasure is the integer ambiguity candidate set â_(k) which provides thelargest residual error norm value, i.e. the largest value for(â−â_(k))^(T)Q_(ã) ⁻¹(â−â_(k)). In one embodiment, the worst qualitymeasure is the one which is the farthest from the best quality measure.

One aspect of the invention further includes an apparatus to estimateparameters derived from GNSS signals useful to determine a position. Theapparatus includes a receiver of a GNSS signal from each of a pluralityof GNSS satellites; a filter having a state vector at least comprising afloat ambiguity for each received frequency of the GNSS signals, eachfloat ambiguity constituting a real number estimate associated with aninteger number of wavelengths of the GNSS signal between a receiver ofthe GNSS signal and the GNSS satellite from which it is received, andthe filter being for estimating a float value for each float ambiguityof the state vector; and a processing element adapted, capable orconfigured to

-   -   assign integer values to at least a subgroup of the estimated        float values to define a plurality of integer ambiguity        candidate sets;    -   select a first number of candidate sets having a quality measure        better than a first threshold, wherein the first threshold is        determined based on a reference quality measure of a reference        candidate set; and    -   obtain a weighted average of the selected candidate sets, each        candidate set weighted in the weighted average based on its        quality measure.

In one embodiment of the apparatus, the processing element is adapted,capable or configured to use the weighted average to estimate a positionof the receiver of the GNSS signals.

In one embodiment of the apparatus, the reference candidate set is thecandidate set having the best quality measure.

In one embodiment of the apparatus, the quality measure of the candidatesets is constituted by a residual error norm value, the residual errornorm value of a candidate set being a measure for a statistical distanceof the candidate set to the state vector having the float ambiguities.

In one embodiment of the apparatus, the processing element is adapted,capable or configured to determine the first threshold as at least oneof a fraction of, a multiple of, and a distance to the reference qualitymeasure.

In one embodiment of the apparatus, the processing element is adapted,capable or configured to, if the number of selected candidate sets(referred to herein as first number of selected candidate sets) issmaller than a second threshold, selecting, form the weighted average, asecond number of further candidate sets on the basis of the qualitymeasures of the candidate sets in decreasing order starting with thenon-selected candidate set having the best quality measure, the secondnumber being constituted by the difference between the first number ofselected candidate sets and a second threshold defining a minimum numberof candidate sets to be included in the weighted average.

In one embodiment of the apparatus, the processing element is adapted,capable or configured to, if the number of selected candidate sets(referred to herein as first number of selected candidate sets) islarger than a third threshold, exclude a third number of selectedcandidate sets from forming the weighted average in decreasing orderstarting with the selected candidate set having the worst qualitymeasure, wherein the third number is constituted by the differencebetween the first number of selected candidate sets and a thirdthreshold defining a maximum number of candidate sets to be included inthe weighted average.

As explained in more details in section 6 entitled “[6. Combination ofaspects and embodiments, and further considerations applicable to theabove]”, the receiver, the filter and the processing element of theabove-described apparatuses may be separate from each other. As alsoexplained in more details in section 6, the invention also relates to acomputer program, to a computer program medium, to a computer programproduct and to a firmware update containing code instructions forcarrying out any one of the above-described methods.

[2. Scaling of Quality Measure]

It has been mentioned above that the selection of candidate sets may bebased on a threshold determined by the quality measure of a referencecandidate set.

According to the one embodiment, the quality measure of the bestcandidate set is taken, in addition thereto or as an alternative, toadapt how the quality measure of the candidate sets is calculated. Inthis embodiment, the function providing the quality measure for acandidate set is a probability function of correctness of thecombination of integers. The quality measure distribution may beadjusted to the expectation of the underlying observation errors. If thebest candidate set is worse than the expectation value of theobservation errors, the quality measure distribution is broadened. Ifthe best candidate set is better than the expectation value, the qualitymeasure distribution may also be narrowed. Only providing for broadeningthe quality measure distribution and not narrowing it is also possible.

This broadened or narrowed distribution of quality measures is then usedto influence the weighted average.

One option is to influence the selection process of candidate sets byscaling the distribution of quality measures, where a broadeneddistribution leads to including more candidate sets into the weightedaverage and a narrowed distribution leads to including fewer candidatesets into the weighted average. Scaling the distribution in this respectrefers to multiplying quality measures by a scaling factor.

Another option is to influence the forming of the weighted average fromthe selected candidate sets. A broadened distribution should lead to anequalizing tendency, wherein the equalizing tendency means that theinfluence of the contribution of candidate sets having different qualitymeasures will tend be closer to one another so that less influence isput on the best quality measure when calculating the weighted average.In contrast, a narrowed distribution should lead to emphasizing thecontribution of candidate sets having good quality measures in theweighted average so that more influence is put on the candidate sethaving the best quality measure when calculating the weighted average.

Both options may be combined.

According to one embodiment, a method for estimating parameters derivedfrom GNSS signals (wherein the operations for scaling the qualitymeasures are carried out) include obtaining observations of a GNSSsignal from each of a plurality of GNSS satellites; feeding theobservations to a filter having a state vector at least comprising afloat ambiguity for each received frequency of the GNSS signals, eachfloat ambiguity constituting a real number estimate associated with aninteger number of wavelengths of the GNSS signal between a receiver ofthe GNSS signal and the GNSS satellite from which it is received, andthe filter being for estimating a float value for each float ambiguityof the state vector; assigning integer values to at least a subgroup ofthe estimated float values to define a plurality of integer ambiguitycandidate sets; determining a quality measure for each of the candidatesets; determining the best quality measure of the candidate sets;determining an expectation value of the candidate set having the bestquality measure; determining an error measure as a ratio of the bestquality measure to the expectation value; adapting the quality measuresof the candidate sets as a function of the error measure; and obtaininga weighted average of a subgroup of the candidate sets on the basis ofthe adapted quality measures, wherein at least one of selecting thesubgroup of the candidate sets and the weighting of each candidate setin the weighted average is based on the adapted quality measure.

The advantages of the method according to the above-mentioned aspect ofthe invention include the fact that the selection of the candidate setsto include into the weighted average is made in accordance with how wellthe statistical characteristics of the observation errors werepredicted.

One embodiment of the method of the invention is illustrated in FIG. 11a. The method 1100 includes a step 1120 of obtaining observations ofGNSS signals from satellites, which is similar to the step 120 asdescribed with reference to FIG. 5 a. The output of the step is a seriesof observations, which correspond the elements of the vector y ofequation system (1) (see background section). Then, the observations arefed into a filter (step 1140). This step is similar to the step 140described with reference to FIG. 5 a. The output of the step 1140 is afloat solution, wherein the ambiguities are estimated as real numbers.The optional step 1140 a of performing a Z-transformation, asillustrated in FIG. 11 a, will be discussed below.

Integer values are then assigned to each one of the float values of thefloat solution to form integer candidate sets (step 1160). Step 1160 issimilar to step 160 described with reference to FIG. 5 a. In step 1180,some of the candidate sets defined and formed in step 1160 are selectedusing a threshold. In this embodiment, the threshold is not necessarilybased on a reference quality measure, and may instead be a thresholdcorresponding to an absolute quality measure value.

In the embodiment illustrated in FIG. 11 a, after the step 1160 ofassigning integer values to the float solution to define candidate sets,a best candidate set is identified. The best candidate set is used instep 1170. In this step, the expectation of the best candidate set isdetermined, and based on the expectation of the best candidate set, thequality measure of the other candidate sets, or the function forcomputing the quality measure of the candidate set, is modified. Namely,if the quality measure of the best candidate set is better thanexpected, the quality measure function is modified so that the number ofselected candidate sets is decreased. In contrast, if the qualitymeasure of the best candidate set is worse than expected, the qualitymeasure of the other candidate sets is modified so as to include morecandidate sets in the weighted average.

As an output of step 1170, the adapted quality measures are provided toinfluence the selection of candidate sets.

Finally, the weighted average of selected candidate sets is formed instep 1200. The iFlex solution is obtained as an output of step 1200. Theoptional step 1200 a of performing a reverse Z-transformation, asillustrated in FIG. 11 a, is discussed below.

In one embodiment (illustrated by the optional step 1140 a in FIG. 11a), after step 1140 but before step 1160, the float solution istransformed to a more orthogonal space, for instance using a Z-transformas disclosed at the end of section 3.3 of the background section. From acomputational standpoint, the Z-transform approach enables toefficiently cover the integer search space. Although the iFlex techniquemay be carried out without the Z-transform, it is advantageous to usethe Z-transform for a practical implementation. Other decorrelationtechniques may be used. If a Z-transformation is carried out on thefloat solution, a reverse Z-transformation should also be performed, asillustrated by the optional step 1200 a in FIG. 11 a. Although notillustrated in all Figures, such as on FIG. 11 b and 11 c, theZ-transformation and the reverse Z-transformation also constituteoptional and advantageous steps in the context of the other embodiments.

FIG. 11 b illustrates another embodiment of a method 1100 of theinvention. In this embodiment, the adapted quality measures are providedto influence the weights of the weighted average of candidate setsformed in step 1200. Step 1180 illustrated in FIG. 11 a is notrepresented in FIG. 11 b because in this case all the candidate setswithin an uncertainty region may be taken into account for forming theweighted average as a basis of the iFlex solution.

FIG. 11 c illustrates a method 1100 of an embodiment of the invention,which constitutes a combination of the embodiments illustrated in FIGS.11 a and 11 b. Namely, the best candidate set is used in step 1170 asfollows. The expectation of the best candidate set is determined and thequality measures are adapted accordingly. The adapted quality measuresare provided to influence both the candidate set selection (in step1180) and the weights of the weighted average (formed in step 1200).

FIG. 12 illustrates more details about the transition between steps 1160and 1170 in FIGS. 11 a to 11 c. After the step 1160 of assigning integervalues to the float solution to define candidate sets, the qualitymeasures of each candidate sets is determined (step 1161). The bestcandidate set is then identified (step 1162) and the expectation of thebest candidate set is determined so as to adapt the quality measures(step 1170). The adapted quality measures, which are the output of step1170, are used for influencing the selection of candidate sets or theweights of the weighted average, or both.

FIGS. 13 a and 13 b illustrate two embodiments of the step 1170consisting in determining the expectation of the best candidate sets andadapting the quality measures accordingly.

In FIG. 13 a, the covariance matrix of the best solution, i.e. of thebest candidate sets, is computed (step 1171). Then, the unit variance iscomputed (step 1172). If the unit variance is smaller than or equal to1.0, no scaling is performed (step 1173). Otherwise, namely if the unitvariance is larger than 1.0, the covariance matrix of the float solutionis scaled (step 1174). If the unit variance is larger than 1.0, thismeans that the statistical properties of the residual were set in a toooptimistic manner. This practical implementation of adapting the qualitynetwork as a function of the expectation of the best candidate set maythen be used to influence the candidate set selection and/or the weightsof the weighted average to form the iFlex solution.

FIG. 13 b illustrates an embodiment of the step 1170 which is similar tothe one described with reference to FIG. 13 a, except that thecovariance matrix of the float solution is scaled (step 1174) not onlywhen the unit variance is larger than 1.0 (meaning that an optimisticchoice of the statistical properties of the residuals has been made),but also when the unit variance is smaller than 1.0 (meaning that apessimistic choice of the statistical properties of the residuals hasbeen made). In other words, if it is determined, in step 1172, that theunit variance is equal to 1.0, no scaling is performed (step 1173). If,in contrast, it is determined in step 1172, that the unit variance isdifferent than 1.0, the covariance matrix of the float solution isscaled (step 1174).

A margin around the value 1.0 may be taken. For instance, step 1172 mayamount to checking whether the unit variance is comprised between1.0−epsilon and 1.0+epsilon, where epsilon is a margin having forinstance a value of 0.05.

The scaling is used to adjust, or change, the statistical properties ofthe noise (residuals e of equation system (1)).

FIG. 14 illustrates on the left-hand side the scaling of the probabilitydistribution of the residuals in an optimistic situation (unit variancebeing larger than 1.0 or larger than (1.0+epsilon)). The scalingillustrated on the right-hand side of FIG. 14 illustrates the scaling ofthe probability distribution of the residuals in the case of apessimistic choice of the statistical properties of the residuals (unitvariance smaller than 1.0 or smaller than (1.0− epsilon)).

A few exemplary candidate sets (denoted c0, c1, c8) are shown asvertical lines from the horizontal axis on the probability distributionfunction (PDF) graphs of FIG. 14. It can be seen that the scalingresults in changes in the weights for the candidate sets (shown asdifferent heights of the vertical lines on the PDF) as well as possiblyincluding more candidate sets or fewer candidate sets. The left-handside of FIG. 14 shows four candidate sets {c0, c1, c2, c3}before scalingof the covariance matrix of the float solution (top left graph on FIG.14), and seven candidate sets {c0, c1, c2, c3, c4, c5, c6}after scalingof the covariance matrix (bottom left graph on FIG. 14). The right-handside of FIG. 14 shows four candidate sets {c0, c1, c2, c3} beforescaling (top right graph on FIG. 14), and two candidate sets {c0, c1}after scaling (bottom right graph on FIG. 14). The graphs are onlyexamples and actual cases may include many more candidate sets.

In one embodiment, the above-described method includes adapting thequality measures of the candidate sets as a function of the errormeasure by adjusting the variance-covariance matrix of the floatsolution.

In one embodiment, the above-described method includes adjusting avariance-covariance matrix of the filter using the error measure, if theerror measure is in a predetermined range.

In one embodiment, the above-described method includes adjusting avariance-covariance matrix of the filter using the error measure, if theerror measure is larger than one.

One aspect of the invention further includes an apparatus to estimateparameters derived from GNSS signals useful to determine a position. Theapparatus includes a receiver adapted to obtain observations of a GNSSsignal from each of a plurality of GNSS satellites; a filter having astate vector at least comprising a float ambiguity for each receivedfrequency of the GNSS signals, each float ambiguity constituting a realnumber estimate associated with an integer number of wavelengths of theGNSS signal between a receiver of the GNSS signal and the GNSS satellitefrom which it is received, and the filter being for estimating a floatvalue for each float ambiguity of the state vector; and a processingelement adapted, capable or configured to

-   -   assign integer values to at least a subgroup of the estimated        float values to define a plurality of integer ambiguity        candidate sets;    -   determine a quality measure for each of the candidate sets;    -   determine the best quality measure of the candidate sets;    -   determine an expectation value of the candidate set having the        best quality measure;    -   determine an error measure as a ratio of the best quality        measure to the expectation value;    -   adapt the quality measures of the candidate sets as a function        of the error measure; and    -   obtain a weighted average of a subgroup of the candidate sets on        the basis of the adapted quality measures, wherein at least one        of selecting the subgroup of the candidate sets and the        weighting of each candidate set in the weighted average is based        on the adapted quality measure.

In one embodiment of the apparatus, the processing element is adapted,capable or configured to adapt the quality measures of the candidatesets as a function of the error measure by adjusting thevariance-covariance matrix of the float solution.

In one embodiment of the apparatus, the processing element is adapted,capable or configured to adjust a variance-covariance matrix of thefilter using the error measure, if the error measure is in apredetermined range.

In one embodiment of the apparatus, the processing element is adapted,capable or configured to adjust a variance-covariance matrix of thefilter using the error measure, if the error measure is larger than one.

As explained in more details in section 6 entitled “[6. Combination ofaspects and embodiments, and further considerations applicable to theabove]”, the receiver, the filter and the processing element of theabove-described apparatuses may be separate from each other. As alsoexplained in more details in section 6, the invention also relates to acomputer program, to a computer program medium, to a computer programproduct and to a firmware update containing code instructions forcarrying out any one of the above-described methods.

[3. Indication of Convergence of Weighted Average Solution]

In one embodiment, a method to estimate parameters derived from GNSSsignals useful to determine a position, includes obtaining observationsof a GNSS signal from each of a plurality of GNSS satellites; feedingthe observations to a filter having a state vector comprising a floatambiguity for each received frequency of the GNSS signals, each floatambiguity constituting a non integer estimate of an integer number ofwavelengths of the GNSS signal between a receiver of the GNSS signal andthe GNSS satellite from which it is received, the filter estimating afloat value for each float ambiguity of the state vector and covariancevalues associated with the state vector; assigning integer values to atleast a subgroup of the estimated float values to define a plurality ofinteger ambiguity candidate sets; obtaining a weighted average of thecandidate sets; determining a formal precision value based on covariancevalues of the filter, the formal precision value being a measure for anachievable precision; determining an achieved precision value of theweighted average; comparing the achieved precision value with the formalprecision value to obtain a convergence value; and indicating aconvergence of the determination of the state vector.

The advantages of this aspect of the invention notably include providingan indication to the users as to which extent the iFlex has converged toa float fixed solution.

In one embodiment of the invention, illustrated in FIG. 15, a furthermethod 2100 to estimate parameters derived from GNSS signals useful todetermine a position is provided. The considerations regarding theZ-transformation, as mentioned above with respect to other embodiments,also apply to the present embodiments.

The method includes a step 2120 of obtaining observations of a GNSSsignal from each of a plurality of GNSS satellites. This step is similarto the steps 120 and 1120 as described with reference to FIG. 5 a andFIG. 11 a to 11 c.

The method further includes a step 2140 of feeding the observations to afilter having a state vector comprising a float ambiguity for eachreceived frequency of the GNSS signals, each float ambiguityconstituting a real number estimate of an integer number of wavelengthsof the GNSS signal between a receiver of the GNSS signal and the GNSSsatellite from which it is received, the filter estimating a float valuefor each float ambiguity of the state vector and covariance valuesassociated with the state vector. This step is similar to the steps 140and 1140 as described with reference to FIG. 5 a and FIG. 11 a to 11 c.

The method further includes a step 2160 of assigning integer values tothe estimated float values, or to at least a subgroup of the estimatedfloat values, to define a plurality of integer ambiguity candidate sets.This step is similar to the steps 160 and 1160 as described withreference to FIG. 5 a and FIG. 11 a to 11 c.

A weighted average of the candidate sets is then outputted to form theso-called iFlex solution (step 2200). All the integer ambiguitycandidate sets of an uncertainty region as illustrated with reference toFIGS. 2 a to 2 e may be selected in step 2160 for forming the weightedaverage in step 2200. Alternatively, only a subset of the candidate setsof an uncertainty region may be selected. Namely, the candidate setsbeing the closest to the float solution (output of step 2140) may beselected for forming the weighted average in step 2200.

In step 2205, a formal precision value is determined based on thecovariance values of a completely fixed-integer solution. That is, fromthe covariance matrix of the float solution obtained from step 2140, theformal precision is determined. This is done by assuming that, in thecovariance matrix of the float solution, the integer values are known.This provides a “formal precision”. The expression “formal precision”here relates to the precision a correct fixed solution would have, whichis the equivalent to a completely converged iFlex solution.

In step 2210, an achieved precision value of the weighted average, i.e.of the iFlex solution, is determined. The achieved precision correspondsto the formal precision plus the additional uncertainty caused by theiFlex weighted sum.

The achieved precision value (obtained in step 2210) is then comparedwith the formal precision value (obtained in step 2205) to obtain aconvergence value (step 2215).

A convergence of the determination of the state vector is then indicated(step 2218). The indication of convergence provides information to theuser(s) (e.g., to the user of a rover or more generally to an observerof the processing operation at a network processing station) as to howclose the iFlex solution has converged with respect to the correct fixedsolution. This is because the achieved precision takes into account theerror due to the iFlex solution (due to averaging the selected candidatesets) and the theoretical margin error of a fixed solution (obtained byfixing the integer values). When the error due to the iFlex solutionapproaches zero, the achieved precision approaches the theoreticalmargin error and the iFlex may be declared fixed to the user.

The formal precision may be obtained as follows. The float solutionobtained in step 2140 may be characterized by the vector:

$\begin{matrix}\begin{bmatrix}\hat{a} \\\hat{b}\end{bmatrix} & (12)\end{matrix}$wherein

-   -   the diacritic symbol ^ above parameters indicates that the        parameters are derived from the float solution;    -   â is the vector of dimension n comprising the carrier phase        ambiguities derived from the float solution; and    -   {circumflex over (b)} is the vector of dimension p comprising        the remaining parameters, i.e. the so-called baseline        parameters, also derived from the float solution (see        equation (1) in background section).

The covariance matrix associated with the float solution is

$\begin{matrix}\begin{bmatrix}Q_{\hat{a}} & Q_{\hat{a}\hat{b}} \\Q_{\hat{b}\hat{a}} & Q_{\hat{b}}\end{bmatrix} & (13)\end{matrix}$Once the float solution has been mapped to an integer solution, bymapping the real number estimated of carrier phase ambiguities fromR^(n) to Z^(n), the formal precision is obtained by the expression:Q_({circumflex over (b)})−Q_({circumflex over (b)}â)Q_(â)⁻¹Q_({circumflex over (b)}â) ^(T)  (14)wherein

-   -   Q_({circumflex over (b)}) is the covariance matrix of the        baseline parameters derived from the float solution;    -   Q_({circumflex over (b)}â) is the covariance matrix of the        baseline parameters derived from the float solution and the        carrier phase ambiguities derived from the float solution;    -   Q_(â) ⁻¹ is the inverse of the covariance matrix of the carrier        phase ambiguities derived from the float solution; and

Q_({circumflex over (b)}â) ^(T) is the transpose of the covariancematrix Q_({circumflex over (b)}â).

The achieved precision may be obtained as follows. The covariance matrixof the iFlex solution is calculated by the expression:

$\begin{matrix}{\left\lbrack {Q_{\hat{b}} - {Q_{\hat{b}\hat{a}}Q_{\hat{a}}^{- 1}Q_{\hat{b}\hat{a}}^{T}}} \right\rbrack + \left\lbrack {\frac{Q_{{\overset{\_}{b}}_{SUM}}}{\kappa_{SUM}} - {{\overset{\_}{b}}_{IFLEX}{\overset{\_}{b}}_{IFLEX}^{T}}} \right\rbrack} & (15)\end{matrix}$wherein

-   -   Q _(b) _(SUM) is the covariance of the accumulated iFlex        baseline parameters, which is equal to

${\sum\limits_{{\hat{a}}_{k} \in Z^{n}}{{\overset{\Cap}{b}}_{k}{\overset{\Cap}{b}}_{k}^{T}{\kappa\left( {\overset{\Cap}{a}}_{k} \right)}}};$

-   -   κ(â_(k)) is the weight for integer ambiguity candidate a â_(k)    -   κ_(SUM) is the sum of weights accumulated from each candidate;    -   b _(IFLEX) is the iFlex baseline parameters; and    -   b _(IFLEX) ^(T) is the transpose of b _(IFLEX).        In other words,

$\left\lbrack {\frac{Q_{{\overset{\_}{b}}_{SUM}}}{\kappa_{SUM}} - {{\overset{\_}{b}}_{IFLEX}{\overset{\_}{b}}_{IFLEX}^{T}}} \right\rbrack$is the contribution to the covariance of the iFlex solution due to theweighted sum, or the additional uncertainty caused by the iFlex weightedsum, on top of the theoretical uncertainty due to the GNSS model itself.

When the trace of

${\left\lbrack {Q_{\hat{b}} - {Q_{\hat{b}\hat{a}}Q_{\hat{a}}^{- 1}Q_{\hat{b}\hat{a}}^{T}}} \right\rbrack + \left\lbrack {\frac{Q_{{\overset{\_}{b}}_{SUM}}}{\kappa_{SUM}} - {{\overset{\_}{b}}_{IFLEX}{\overset{\_}{b}}_{IFLEX}^{T}}} \right\rbrack},$which is the achieved precision, approaches the trace of[Q_({circumflex over (b)})−Q_({circumflex over (b)}â)Q_(â)⁻¹Q_({circumflex over (b)}â) ^(T)], which is the formal precision, theiFlex position covariance matrix approaches the position covariancematrix of the integer ambiguity solution (i.e. the integer-constrainedambiguity solution). The degree of convergence may therefore beindicated by comparing the achieved precision and the formal precision.

When it is determined in the comparison step 2215 that the achievedprecision is sufficiently close to the formal precision, an indicationof convergence value is provided to the user, indicating that the iFlexsolution has converged sufficiently as to be practically equivalent to acorrect fixed solution for the user's purposes.

In one embodiment, the convergence value is obtained in step 2215 bycomparing the achieved precision value with the formal precision value,wherein comparing includes computing the ratio of the achieved precisionvalue to the formal precision value.

The ratio computed to obtain the convergence value may for instance be:

$\begin{matrix}\frac{{tr}\left\{ {\left\lbrack {Q_{\hat{b}} - {Q_{\hat{b}\hat{a}}Q_{\hat{a}}^{- 1}Q_{\hat{b}\hat{a}}^{T}}} \right\rbrack + \left\lbrack {\frac{Q_{{\overset{\_}{b}}_{SUM}}}{\kappa_{SUM}} - {{\overset{\_}{b}}_{IFLEX}{\overset{\_}{b}}_{IFLEX}^{T}}} \right\rbrack} \right\}}{{tr}\left\{ \left\lbrack {Q_{\hat{b}} - {Q_{\hat{b}\hat{a}}Q_{\hat{a}}^{- 1}Q_{\hat{b}\hat{a}}^{T}}} \right\rbrack \right\}} & (16)\end{matrix}$Wherein the “tr{ }” operator is the trace operator (the trace of asquare matrix is the sum of the elements on the main diagonal, i.e. thediagonal from the upper left to the lower right).

Alternatively, the ratio computed to obtain the convergence value mayfor instance be:

$\begin{matrix}\frac{\begin{matrix}{\det\left\{ {\left\lbrack {Q_{\hat{b}} - {Q_{\hat{b}\hat{a}}Q_{\hat{a}}^{- 1}Q_{\hat{b}\hat{a}}^{T}}} \right\rbrack +} \right.} \\\left. \left\lbrack {\frac{Q_{{\overset{\_}{b}}_{SUM}}}{\kappa_{SUM}} - {{\overset{\_}{b}}_{IFLEX}{\overset{\_}{b}}_{IFLEX}^{T}}} \right\rbrack \right\}\end{matrix}}{\det\left\{ \left\lbrack {Q_{\hat{b}} - {Q_{\hat{b}\hat{a}}Q_{\hat{a}}^{- 1}Q_{\hat{b}\hat{a}}^{T}}} \right\rbrack \right\}} & (17)\end{matrix}$wherein the det{ } operator is the matrix determinant operator.

In one embodiment, the method described with reference to FIG. 16 afurther includes a step of determining 2225 an instance in time when theachieved precision of the position is better than a convergencethreshold; and a step of indicating 2230 a convergence of thedetermination of the state vector at and after the determined instancein time.

This will be better understood with reference to FIG. 16 a, which is aflowchart of one embodiment of the step 2218 of providing an indicationof a convergence value, as illustrated in FIG. 15. Step 2218 comprises asubstep 2225 of determining that the convergence (which may be the ratioof the achieved precision value to the formal precision value, asmentioned above) is better than a convergence threshold at an instantt=t0. After said determination is made in step 2225, an absolute orfixed convergence is indicated in step 2230 from the instant t0 to theinstant t0+ΔH, wherein ΔH is a duration during which the indication offixed convergence is kept despite the convergence value being below theconvergence threshold.

This mechanism provides retention of the fixed indication to theuser(s). Flickering of the convergence indication between an indication“fixed” and an indication “float” is therefore avoided.

The duration ΔH may be constant (and for instance determined in advanceas a configuration parameter, i.e. predetermined), as illustrated inFIGS. 16 b and 16 c, or may depend on the extent of deviation of theconvergence value below the convergence threshold, as illustrated inFIGS. 16 d and 16 e.

Examples of implementation of step 2218 of providing a convergence valueindication will now be described with reference to FIGS. 16 b to 16 e.

FIG. 16 b illustrates the evolution in time of the convergence of theiFlex solution with respect to the correct fixed solution. As mentionedabove, the convergence may be computed as the ratio of the achievedprecision to the formal precision. A convergence value of 1 (one) wouldmean that the weighted average solution has converged completely to thefixed solution, and that any further improvement cannot be expected fromthe weighted average of selected candidate sets.

Starting on the left-hand side of the graph of FIG. 16 b, theconvergence value is shown to increase, while still being below theconvergence threshold (having a value of 0.95 for instance). In thissituation, the indication provided to the user is “float” (asillustrated on the bottom of FIG. 16 b). Once the convergence valuereaches the convergence threshold, the indication provided to the useron the receiver is switched from “float” to “fixed”. As long as theconvergence value stays above the convergence threshold, the convergenceindication is “fixed”. Thereafter, when the convergence value decreasesbelow the convergence threshold, the provided convergence indication isnot immediately switched to “float”. During a duration ΔH(t) duringwhich the convergence value is below the convergence threshold, theindication stays “fixed”. The duration ΔH(t) is the so-called indicationhysteresis.

The instant t0, described with reference to step 2225 in FIG. 16 a,corresponds the point in time wherein the convergence value crosses theconvergence thresholds downwards.

Once the duration ΔH(t) is exceeded, i.e. once the convergence value hasbeen below the convergence threshold for the duration longer than ΔH(t),the convergence indication is switched from “fixed” to “float”.

Finally, still as shown on FIG. 16 b, when the convergence value returnsabove the convergence threshold, the indication is switched again from“float” to “fixed”.

FIG. 16 c illustrates a similar example, wherein the hysteresis durationΔH(t) is a constant duration (such as a constant predeterminedduration). In the example illustrated in FIG. 16 c, after theconvergence value has reached the convergence threshold for the firsttime, it does not return back below the convergence threshold fordurations longer than ΔH(t). Therefore, the convergence indication stays“fixed” after first crossing the convergence threshold.

FIG. 16 d illustrates an example wherein the hysteresis duration is notset as a constant duration, but instead the hysteresis duration isdependent on the actual deviation Δc of the convergence value below theconvergence threshold. The notation ΔH(c) indicates that the hysteresisis dependent on the deviation of the convergence value from theconvergence threshold (i.e. below the convergence threshold).

From the left-hand side of FIG. 16 d, it can be shown that theconvergence value increases up to reaching the convergence threshold. Atthat point, the convergence indication is switched from “float” to“fixed”. Then, after staying above the convergence threshold for awhile, the convergence value decreases so as to cross the convergencethreshold and then decreases again below the convergence threshold, to acertain extent.

The convergence indication stays “fixed” during the hysteresis durationΔH(c), i.e. until the convergence value has decreased by the convergencedeviation Δc below the convergence threshold. At that point, theconvergence indication is switched back from “fixed” to “float”.

The indication “float” is kept until the convergence value crosses theconvergence threshold again. At that point, the indication is switchedto “fixed” again.

FIG. 16 e illustrates another example wherein the hysteresis durationdepends on a deviation Δc from the convergence threshold. In the exampleillustrated in FIG. 16 e, after first reaching the convergencethreshold, the convergence value does not decrease below the convergencethreshold by more than the deviation Δc. Therefore, the providedconvergence indication stays “fixed”.

A convergence indication process having hysteresis characteristicsimproves stability and the system usability.

The hysteresis duration has been described above as either having aconstant duration (FIGS. 16 b and c) or a duration dependent on theextent of deviation from the convergence threshold (FIGS. 16 d and e).In one embodiment, the hysteresis duration is dependent both on the timespent by the convergence value under the convergence threshold and onthe extent of deviation from the convergence threshold.

FIG. 17 a illustrates one embodiment of the method described withreference to FIG. 15. In the embodiment illustrated in FIG. 17 a, oncethe achieved precision has been determined (step 2210, not illustratedin FIG. 17 a, but on FIG. 15), it is determined (step 2211) whether theachieved precision is better than an unconditional inclusion threshold.If so, it is indicated (step 2219) that the iFlex solution has converged(“fixed” indication). Otherwise, the steps 2215 and 2218 of comparingthe achieved precision with the formal precision and indicating aconvergence value accordingly are carried out. These steps have beendescribed with reference to FIG. 15.

FIG. 17 b illustrates one embodiment wherein, after the achievedprecision has been determined (step 2210, not illustrated in FIG. 17 a,but on FIG. 15), it is determined (step 2212) whether the achievedprecision is worse than an unconditional exclusion threshold. If so, itis indicated (step 2221) that the iFlex solution has not converged(“float” indication). Otherwise, the steps 2215 and 2218 are carriedout, as described with reference to FIG. 15.

The embodiments described with reference to FIG. 17 a (use of anunconditional inclusion threshold for the achieved precision) and withreference to FIG. 17 b (use of an unconditional exclusion threshold forthe achieved precision) may be combined.

FIGS. 18 a to 18 d illustrate two examples of applying inclusion andexclusion thresholds to the achieved precision determined in step 2210.Namely, FIG. 18 a illustrates an example of the application of theunconditional inclusion threshold, FIG. 18 b illustrates an example ofapplication of the unconditional exclusion threshold, FIG. 18 cillustrates an example of combined application of the unconditionalinclusion threshold and the unconditional exclusion threshold, and FIG.18 d illustrates an example of combined application of the unconditionalinclusion threshold and the unconditional exclusion threshold, whereinhysteresis are applied in relation to both the unconditional inclusionand exclusion thresholds (to improve the indication stability).

FIG. 18 a illustrates an example of evolution of the achieved precision.The graph shows the achieved position precision (for instance incentimeters) as a function of time (for instance in seconds). Startingfrom the left-hand side of the graph, as long as the achieved precisionis above the unconditional inclusion threshold, the convergenceindication provided to the user is determined depending on theconvergence value (corresponding for instance to the ratio of theachieved precision value to the formal precision value), as explainedwith reference to FIGS. 16 a to 16 e. As soon as the achieved precisionreaches the unconditional inclusion threshold, the convergenceindication is “fixed”, whatever the value of the formal precision.

FIG. 18 b illustrates an example of application of the unconditionalexclusion threshold. Starting on the left-hand side of the graph, theachieved precision is shown to improve (to decrease on the graph), whilestill being above the unconditional exclusion threshold. In that case(extreme left-hand side of the graph), the provided convergenceindication is “float” (as shown on the bottom of FIG. 18 b) whatever thevalue of the formal precision.

As soon as the achieved precision decreases below the unconditionalexclusion threshold, the convergence indication becomes dependent on theconvergence value, as explained with reference to FIGS. 16 a to 16 e.If, at one point in time, the achieved precision returns to a levelabove the unconditional exclusion threshold, the convergence indicationis “float” whatever the value of the formal precision.

FIG. 18 c illustrates an example of combined application of theunconditional exclusion threshold and the unconditional inclusionthreshold. First, the value of the achieved position precision is abovethe unconditional exclusion threshold so that the indication provided tothe users is “float” whatever the value of the formal precision (extremeleft-hand side of the graph). When the achieved precision is comprisedbetween the unconditional exclusion threshold and the unconditionalinclusion threshold, the convergence indication becomes dependent on theconvergence value, as explained with reference to FIGS. 16 a to 16 e.When the achieved position precision decreases below the unconditionalinclusion threshold, the convergence indication is “fixed” whatever thevalue of the formal precision.

FIG. 18 d illustrates the combined application of the unconditionalinclusion threshold and the unconditional exclusion threshold to theachieved precision, wherein the two thresholds are subject to hysteresis(to improve the stability of the indications provided to the users). Theexample of FIG. 18 d is similar to the example of FIG. 18 c except thatwhen the achieved position precision increases above the unconditionalexclusion threshold, the unconditional exclusion threshold is decreasedby a certain deviation (this results in the application of ahysteresis). The achieved precision then only leaves the exclusionregion (wherein the convergence indication “float” is provided whateverthe value of the formal position), if the achieved precision decreasesbelow the temporarily decreased unconditional exclusion threshold. Assoon as the achieved precision has decreased below the temporarilydecreased unconditional exclusion threshold, the unconditional exclusionthreshold is set back to its original level.

The same applies to the unconditional inclusion threshold. Namely, assoon as the achieved precision decreases below the unconditionalinclusion threshold, the level of the unconditional inclusion thresholdis raised by a given deviation. The indication “fixed” is provided tothe users constantly and whatever the value of the formal precisionuntil the achieved precision increases above the temporarily increasedunconditional inclusion threshold. At that point in time, theunconditional inclusion threshold is set back to its original level.

In one embodiment of the method, the convergence value is obtained as aratio of the achieved precision value to the formal precision value.

In one embodiment, the method includes determining an instance in timewhen the convergence value of the position is better than a convergencethreshold; and indicating a convergence of the determination of thestate vector at and after the determined instance in time.

In one embodiment, the method includes estimating an achieved precisionof a position of the receiver determined based on the weightedambiguities; and indicating a convergence of the determination of thestate vector, if the achieved precision of the position is better thanan inclusion threshold.

In one embodiment, the method includes estimating an achieved precisionof a position of the receiver determined based on the weightedambiguities; and indicating a non-convergence of the determination ofthe state vector, if the achieved precision of the position is worsethan an exclusion threshold.

One aspect of the invention further includes an apparatus to estimateparameters derived from GNSS signals useful to determine a position,including a receiver adapted to obtain observations of a GNSS signalfrom each of a plurality of GNSS satellites; a filter having a statevector comprising a float ambiguity for each received frequency of theGNSS signals, each float ambiguity constituting a non integer estimateof an integer number of wavelengths of the GNSS signal between areceiver of the GNSS signal and the GNSS satellite from which it isreceived, the filter estimating a float value for each float ambiguityof the state vector and covariance values associated with the statevector; and a processing element adapted, capable or configured to

-   -   assign integer values to at least a subgroup of the estimated        float values to define a plurality of integer ambiguity        candidate sets;    -   obtain a weighted average of the candidate sets;    -   determine a formal precision value based on covariance values of        the filter, the formal precision value being a measure for an        achievable precision;    -   determine an achieved precision value of the weighted average;    -   compare the achieved precision value with the formal precision        value to obtain a convergence value; and    -   indicate a convergence of the determination of the state vector.

In one aspect of the apparatus, the processing element is adapted,capable or configured to obtain the convergence value as a ratio of theachieved precision value to the formal precision value.

In one aspect of the apparatus, the processing element is adapted,capable or configured to determine an instance in time when theconvergence value of the position is better than a convergencethreshold; and indicate a convergence of the determination of the statevector at and after the determined instance in time.

In one aspect of the apparatus, wherein the processing element isadapted, capable or configured to estimate an achieved precision of aposition of the receiver determined based on the weighted ambiguities;and indicate a convergence of the determination of the state vector, ifthe achieved precision of the position is better than an inclusionthreshold.

In one aspect of the apparatus, the processing element is adapted,capable or configured to estimate an achieved precision of a position ofthe receiver determined based on the weighted ambiguities; and indicatea non-convergence of the determination of the state vector, if theachieved precision of the position is worse than an exclusion threshold.

As explained in more details in section 6 entitled “[6. Combination ofaspects and embodiments, and further considerations applicable to theabove]”, the receiver, the filter and the processing element of theabove-described apparatuses may be separate from each other. As alsoexplained in more details in section 6, the invention also relates to acomputer program, to a computer program medium, to a computer programproduct and to a firmware update containing code instructions forcarrying out any one of the above-described methods.

[4. Keeping Legacy Observations after Interruption of Tracking]

For determining the carrier phase ambiguities, i.e. the unknown numberof cycles of the observed carrier signal from each of the satellites tothe receiver, a filter is used. The filter estimates the state of thedynamic system (the GNSS system). The state is represented by a statevector including the carrier phase ambiguities (undifferenced orsingle-differenced or double-differenced ambiguities).

The ambiguities indicate the unknown number of cycles of the carrierbetween the receiver and the satellite at a particular instance in time,e.g. when initializing the system, and thus are fixed values. Bycollecting more and more observations into the filter, the state vectorof the filter gradually converges to stable values of the ambiguities.

During the evolving time, usually a phase locked loop tracks the carriersignal and determines the additional number of cycles to be added ordeducted from the initial value that is to be estimated by the filter.In principle, the distance between the receiver and the satellite canthen be determined at a particular instance in time on the basis of theinitial unknown number of cycles of the state vector and theadditionally tracked number of oscillations of the phase locked loop.The distance to multiple satellites determines the position of thereceiver. Single-differenced or double-differenced ambiguities betweendifferent satellites and receivers, however, are also useful fordetermining a position of the receiver.

When a satellite rises or sets or otherwise is obstructed and thecorresponding observations of a signal cannot or only insufficiently bemade, in other words, the signal cannot be tracked, the ambiguity valuecorresponding to this signal of the satellite conventionally may beremoved from the state vector. If the satellite reappears or if a newsatellite appears, ambiguities may then be added to the state vector andupdated over subsequent instances in time.

In an alternative, in one embodiment of the invention, rather thanremoving ambiguities for frequencies that drop out and addingambiguities for appearing frequencies, an ambiguity value of a signal inthe state vector is maintained despite the loss of tracking of thesignal. It has been recognized that even if the GNSS signal cannot betracked, the accuracy of the weighted average determined on the basis ofthe candidate sets may be improved.

It may even be considered to maintain the previous last trackedambiguity value after an interruption in tracking for a certain periodin time, before allowing an update of this ambiguity value in the statevector, in order to allow first collecting enough observations.

In one embodiment, a method is provided to estimate parameters derivedfrom GNSS signals useful to determine a position, including obtainingobservations of each of received frequencies of a GNSS signal from aplurality of GNSS satellites to obtain observations at a plurality ofinstances in time; feeding the time sequence of observations to a filterto estimate a state vector at least comprising float ambiguities,wherein each float ambiguity constitutes a non integer estimate of aninteger number of wavelengths for a received frequency of a GNSS signalbetween a receiver of the GNSS signal and the GNSS satellite from whichit is received and wherein the float ambiguities of the state vector areupdated over time on the basis of the observations; determining that aninterruption in tracking of at least one signal of a satellite occurred;maintaining the float ambiguity of the state vector for the at least onesignal for which an interruption in tracking occurred at the valuebefore the interruption in tracking occurred; assigning integer valuesto at least a subgroup of the estimated float values to define aplurality of integer ambiguity candidate sets; determining a qualitymeasure for each of the candidate sets; and obtaining a weighted averageof the candidate sets.

The advantages of this aspect of the invention notably include the factthat the precision of the float solution, and therefore also of theweighted average based therein is improved. This is because more data istaken into account in the filter, and more variables are maintained inthe state vector.

In one embodiment, a method is provided to estimate parameters derivedfrom GNSS signals useful to determine the position. The considerationsregarding the Z-transformation, as mentioned above with respect to otherembodiments, also apply to the present embodiments. As illustrated inFIG. 19, the method 3100 includes a step 3120 of obtaining observationsof each of the received frequencies of a GNSS signal from a plurality ofGNSS satellites to obtain observations at the plurality of instances intime. This step corresponds to the step 120 of FIG. 5 a, the step 1120of FIG. 11 a, and the step 2120 of FIG. 15, except that it is herewithspecified that the observation of the frequencies of the GNSS signalsare obtained at a plurality of instance in time.

The observations at the plurality of instances in time are fed in step3140 into a filter to estimate a state vector at least comprising floatambiguities. Each float ambiguity constitutes a real number estimate ofan integer number of wavelengths for a received frequency of a GNSSsignal between a receiver and a satellite. The float ambiguities of thestate vector are updated over time on the basis of the observations. Thefilter may be implemented using a recursive filter for estimating thestate of a dynamic system (in the present case, the GNSS system), such aKalman filter.

The method also comprises a step 3121 of determining that aninterruption in tracking of at least one signal of a satellite hasoccurred. This may for instance be caused by a satellite being no longerobserved or observable (for instance because it has set).

The method also includes a step 3122 consisting in maintaining the floatambiguity of the state vector for the at least one signal for which aninterruption in tracking has occurred. Float ambiguity of the statevector is maintained at the value it was before the interruption intracking.

The steps 3140 and 3122 both relate to the operation of the filter whichprovides a float solution, which has been already described withreference to FIG. 5 a.

The method further includes a step 3160 of assigning integer values toat least a subgroup of the estimated float values to define theplurality of integer ambiguity candidate sets. Step 3160 corresponds tostep 160 described above.

A quality measure for each of the integer ambiguity candidate sets isthen determined, and a weighted average of the candidate sets isobtained in step 3200. The weighted average provides the iFlex solution,as described with reference to step 200 of FIG. 5 a.

In one embodiment, it is determined that an interruption in tracking ofat least one signal of a satellite has occurred if one observation forat least one signal is not available for at least one of the instance intime (for instance at least one of the GPS epochs).

In one embodiment, it is determined that an interruption in tracking ofat least one signal of a satellite occurred, if a cycle slip hasoccurred. A cycle slip results from a loss of lock of the carrier phasetracking loop of one carrier transmitted from one satellite. A cycleslip causes a discontinuity in tracking of the cycles.

In one embodiment, if, after an interruption in tracking of a signal,the tracking of the signal resumes, the float ambiguity of the statevector for the signal for which an interruption in tracking has occurredis maintained at the value it had before the interruption in trackinghas occurred. This float ambiguity is maintained as a first floatambiguity. In addition, a second float ambiguity is introduced into thestate vector of the filter for taking into account the signal obtainedafter resumption of the tracking. This embodiment will be explained inmore details with reference to FIG. 20.

FIG. 20 includes all the steps described with reference to FIG. 19,except that two steps are added, namely a step 3123 of determining thatan interrupted signal has resumed and a step 3124 of maintaining thefloat ambiguity of the interrupted signal into the state vector of thefilter (as a first float ambiguity) and adding one float ambiguity (asecond float ambiguity) for the resumed signal. This technique improvesthe precision of the provided float solution. Steps 3140, 3122, and 3124relate to the operation of the filter and together output the floatsolution.

In one embodiment of the method, it is determined that an interruptionin tracking of at least one signal of a satellite occurred if anobservation for the at least one signal is not available for at leastone of the instances in time.

In one embodiment of the method, it is determined that an interruptionin tracking of at least one signal of a satellite occurred, if a cycleslip occurred.

In one embodiment, the method includes, if after an interruption intracking of a signal tracking of the signal resumes, maintaining thefloat ambiguity of the state vector for the signal for which aninterruption in tracking occurred at the value before the interruptionin tracking occurred as a first float ambiguity and introducing into thestate vector a second float ambiguity for the signal after resumingtracking.

One aspect of the invention further includes an apparatus to estimateparameters derived from GNSS signals useful to determine a position,including a receiver adapted to obtain observations of each of receivedfrequencies of a GNSS signal from a plurality of GNSS satellites toobtain observations at a plurality of instances in time; a filter toestimate a state vector at least comprising float ambiguities based onthe time sequence of observations, wherein each float ambiguityconstitutes a non integer estimate of an integer number of wavelengthsfor a received frequency of a GNSS signal between a receiver of the GNSSsignal and the GNSS satellite from which it is received and wherein thefloat ambiguities of the state vector are updated over time on the basisof the observations; and a processing element adapted, capable orconfigured to

-   -   determine that an interruption in tracking of at least one        signal of a satellite occurred;    -   maintain the float ambiguity of the state vector for the at        least one signal for which an interruption in tracking occurred        at the value before the interruption in tracking occurred;    -   assign integer values to at least a subgroup of the estimated        float values to define a plurality of integer ambiguity        candidate sets;    -   determine a quality measure for each of the candidate sets; and    -   obtain a weighted average of the candidate sets.

In one aspect of the apparatus, the processing element is adapted,capable or configured to determine that an interruption in tracking ofat least one signal of a satellite occurred if an observation for the atleast one signal is not available for at least one of the instances intime.

In one aspect of the apparatus, the processing element is adapted,capable or configured to determine that an interruption in tracking ofat least one signal of a satellite occurred, if a cycle slip occurred.

In one aspect of the apparatus, the processing element is adapted,capable or configured to, if after an interruption in tracking of asignal tracking of the signal resumes, maintain the float ambiguity ofthe state vector for the signal for which an interruption in trackingoccurred at the value before the interruption in tracking occurred as afirst float ambiguity and introducing into the state vector a secondfloat ambiguity for the signal after resuming tracking.

As explained in more details in section 6 entitled “[6. Combination ofaspects and embodiments, and further considerations applicable to theabove]”, the receiver, the filter and the processing element of theabove-described apparatuses may be separate from each other. As alsoexplained in more details in section 6, the invention also relates to acomputer program, to a computer program medium, to a computer programproduct and to a firmware update containing code instructions forcarrying out any one of the above-described methods.

[5. Ambiguity Selection]

The techniques described herein of forming a weighted average of integerambiguity candidate sets (“iFlex”) have proven to be more robust againstcompromised data than traditional RTK approaches in which theambiguities are fixed and validated. One example is improved performanceunder tree canopy.

However, if there are systematic errors in the data that are notreflected in the variances reported by the floating solution (this partof which is the same for the weighted averaging processes describedherein as for the traditional RTK processing) there is some chance thatthe solution does not converge. The consequence can be the inability todeclare a solution as “fixed” for an extended time period (“no fix”), orworse, solutions reported as having a high precision contain errorslarger than the errors reported by the traditional RTK processing engine(“bad fixes”).

In one such case, caused by an error in the receiver tracking loops, theGLONASS carrier phase data occasionally contained % cycle biases on oneor more satellites. This resulted in extended periods without a precise“fixed” RTK solution, though the traditional RTK system operatedacceptably in those cases by applying the partial fixing technique forGLONASS described in U.S. Pat. Nos. 7,312,747 B2 and 7,538,721 B2.Generalized partial fixing techniques are described in WO 2009/058213A2.

In general, processing of GLONASS data is more problematic thanprocessing GPS data even when the receiver tracking loops are operatingcorrectly: acquisition transients are more frequent for GLONASS datathan for GPS data; and the broadcast orbits for GLONASS are much worsethan GPS broadcast orbits. While the latter are typically accurate atthe 1-2 meter level, GLONASS orbits have errors of 5-10 meters and cansometimes be much worse. This introduces errors on longer baselines. A10 meter orbit error results in a measurement error of up to ½ ppm. Thismeans having an additional error of 35 millimeters on a 70 km baseline,a value already critical for narrow-lane carrier phase ambiguityresolution. In GPS this error would be about 4-8 millimeters. Finally,the L2P-code for GLONASS is neither published nor guaranteed to beunchanged. Any change in the code could lead to cycle slips andacquisition errors.

Another aspect favoring a partial ambiguity resolution scheme is usageof satellite signals that are not currently available and thus nottestable at this time, such as GPS L5 and Galileo signals. If a newsignal as broadcast causes problems in RTK processing, partial ambiguityresolution in accordance with embodiments of the invention could avoidthe need for the user to manually configure the receiver to stop usingthe new signals. Thus, a partial ambiguity resolution scheme is usefulin forming a weighted average of integer ambiguity candidate sets.

Partial search techniques described in U.S. Pat. Nos. 7,312,747 B2 and7,538,721 B2 and in WO 2009/058213 A2 are based around using a fullsearch of partial sets of fixed ambiguities in order to decide whichset(s) can be resolved. Especially for forming a weighted average ofinteger ambiguity candidate sets (the “iFlex” technique) as describedherein this approach can be very computationally intensive.

The following description presents methods and apparatus which canrelatively quickly determine a partial ambiguity set that issignificantly better than the full set of ambiguities, before thecomplete ambiguity solution is evaluated.

Some embodiments of the invention are based on the norm of the bestcandidate solution for each ambiguity subset analyzed. The search forthe best candidate set only is very quick. Though there is nomathematical optimality in this approach, practical tests have shownthat it can improve processing of problematic GNSS observation data.

In other embodiments, the best norm is used as a criterion to detectproblems in the solution. Then the float solution is completely orpartially reset based on the norms of the partial ambiguity sets.

Verhagen, The GNSS integer ambiguities: estimation and validation, DelftUniversity of Technology, 2004, ISBN 90-804147-4-3, provides a goodoverview of the steps of integer ambiguity estimation. Using thenotation established in Section 3 above, a general model of GNSSobservations is:y=Aa+Bb+e  (1)

Recapping from Section 3 that yεR^(m) denotes the observation vector,aεZ^(n) is the vector of unknown integer parameters (ambiguities),bεR^(p) is the vector of additional model parameters as position, clockerrors, atmospheric errors, time correlated noise, etc. A and B are them×n respective m×p design matrices (Jacobians) of the observations withrespect to the integer respective additional parameters. eεR^(m) is theobservation noise vector with the a priori variance-covariance matrixQ=E[ee^(T)].

Normally using a least-squares adjustment, a Kalman filter or any othertechnique known to someone skilled in the art, the unknowns a and b areestimated simultaneously without applying the integer constraint on a.

The result is the so-called float solution of estimates and theirvariance-covariance matrix:

$\begin{matrix}{\begin{pmatrix}\hat{a} \\\hat{b}\end{pmatrix}\mspace{14mu}{and}\mspace{14mu}\begin{pmatrix}Q_{\hat{a}} & Q_{\hat{a}\hat{b}} \\Q_{\hat{b}\hat{a}}^{T} & Q_{\hat{b}}\end{pmatrix}} & \left( {{12\;\&}\mspace{14mu} 13} \right)\end{matrix}$

A least squares solution would be:

$\begin{matrix}{\begin{pmatrix}Q_{\hat{a}} & Q_{\hat{a}\hat{b}} \\Q_{\hat{a}\hat{b}}^{T} & Q_{\hat{b}}\end{pmatrix} = \left( {\begin{pmatrix}A & B\end{pmatrix}^{T}{Q^{- 1}\begin{pmatrix}A & B\end{pmatrix}}} \right)^{- 1}} & (18) \\{\begin{pmatrix}\hat{a} \\\hat{b}\end{pmatrix} = {\begin{pmatrix}Q_{\hat{a}} & Q_{\hat{a}\hat{b}} \\Q_{\hat{a}\hat{b}}^{T} & Q_{\hat{b}}\end{pmatrix}\begin{pmatrix}A & B\end{pmatrix}^{T}Q^{- 1}y}} & (19)\end{matrix}$

This technique is given as an example of float parameter estimation andnot limiting the application of the methods presented.

The variance-covariance matrix Q_(â) defines a norm ∥.∥_(Q) _(â) onR^(n):∥x∥ _(Q) _(â) =x ^(T) Q _(â) ⁻¹ x  (20)

Integer Least-Squares (ILS) is the strict mathematical integer solutionto the problem of minimizing the Q_(â) norm of the difference of thefloat solution to integer solutions:

$\begin{matrix}{{\overset{\Cup}{a}}_{1} = {{{\underset{z \in Z^{n}}{argmin}\left( {z - \hat{a}} \right)}^{T}{Q_{\hat{a}}^{- 1}\left( {z - \hat{a}} \right)}} = {\underset{z \in Z^{n}}{argmin}{{z - \hat{a}}}_{Q_{\hat{a}}}}}} & (21)\end{matrix}$

Well-known algorithms, for example LAMBDA [Teunissen], compute at leastthe “best candidate” {hacek over (a)}₁. In addition, usually additionalnext candidate {hacek over (a)}₂ (“second candidate”), and possible morecandidates {hacek over (a)}₃ . . . , {hacek over (a)}_(k) in order oftheir Q_(â) norm are determined, in a similar way:

$\begin{matrix}{{\overset{\Cup}{a}}_{2} = {\underset{z \in {Z^{n}\backslash{\{\overset{\Cup}{a}\}}}}{argmin}{{z - \hat{a}}}_{Q_{\hat{a}}}}} & (22) \\{{\overset{\Cup}{a}}_{i} = {\underset{z \in {Z^{n}\backslash{\{{{\overset{\Cup}{a}}_{1},\mspace{14mu}\ldots\mspace{14mu},{\overset{\Cup}{a}}_{i - 1}}\}}}}{argmin}{{z - \hat{a}}}_{Q_{\hat{a}}}}} & (23)\end{matrix}$

The values of the norm ∥z−â∥_(Q) _(â) are normally provided, too, forapplication of the validations tests described in the following section.

For a given best candidate {hacek over (a)}_(i) for the integersolution, the non-integer parameters of the model can be easily adjustedto the matching least-squares solution:{hacek over (b)}={circumflex over (b)}−Q _(â{circumflex over (b)}) ^(T)Q _(â) ⁻¹(â−{hacek over (a)} _(i))  (24)andQ _({hacek over (b)}) =Q _({circumflex over (b)}) −Q_(â{circumflex over (b)}) ^(T) Q _(â) ⁻¹ Q_(â{circumflex over (b)})  (25)

Partial fixing refers to selecting a subset of the ambiguitiesdetermined by computing a floating solution in order to get a result(with fixed-integer ambiguities or with a weighted average of integerambiguity candidate sets) that is better than using the completeambiguity set.

In a more general sense, the full ambiguity set is transformed into alower dimensional one using an integer transformation matrix, asdescribed in WO 2009/058213 A2. In a generalized sense, partially fixingambiguities corresponds to fixing a transformed ambiguity vector using arectangular transformation matrix. Here, a (n_(f)×n) matrix G maps theambiguity vector to the part that is going to be fixed. For example,given a floating solution with 6 ambiguities

$\begin{matrix}{\hat{a} = \begin{pmatrix}{\hat{a}}_{1} \\{\hat{a}}_{2} \\{\hat{a}}_{3} \\{\hat{a}}_{4} \\{\hat{a}}_{5} \\{\hat{a}}_{6}\end{pmatrix}} & (26)\end{matrix}$and fixing only the first and last two, G would be:

$\begin{matrix}{{G_{\{{1,2,5,6}\}} = \begin{pmatrix}1 & 0^{I} & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}}{and}} & (27) \\{{G_{\{{1,2,5,6}\}} \cdot \overset{\Cap}{a}} = \begin{pmatrix}{\overset{\Cap}{a}}_{1} \\{\overset{\Cap}{a}}_{2} \\{\overset{\Cap}{a}}_{5} \\{\overset{\Cap}{a}}_{6}\end{pmatrix}} & (28)\end{matrix}$

If for six ambiguities, where only the wide-lane combinations are to befixed, G would be:

$\begin{matrix}{{G_{WL} = \begin{pmatrix}1 & {- 1} & 0 & 0 & 0 & 0 \\0 & 0 & 1 & {- 1} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & {- 1}\end{pmatrix}}{and}} & (29) \\{{G_{WL} \cdot \overset{\Cap}{a}} = \begin{pmatrix}{{\overset{\Cap}{a}}_{1} - {\overset{\Cap}{a}}_{2}} \\{{\overset{\Cap}{a}}_{3} - {\overset{\Cap}{a}}_{4}} \\{{\overset{\Cap}{a}}_{5} - {\overset{\Cap}{a}}_{6}}\end{pmatrix}} & (30)\end{matrix}$

Note that the wide-lane combination is formed by subtracting twoambiguities (or phase measurements) observed to the same satellite atthe same epoch on different frequency bands. For example, the GPS L1/L2wide-lane combination is formed by subtracting the L2 phase from the L1phase. The L1 and L2 phase measurements have wavelengths of ˜19 and 24cm respectively, while the corresponding wide-lane combination has aneffective wavelength of ˜86 cm. For the purposes of ambiguityresolution, it is advantageous to have a wavelength as long as possible,hence the motivation to consider the use of wide-lane phase/ambiguitiesin partial fixing.

There are many other useful linear combinations of multi-band GNSS,including:

-   -   Iono-free,    -   Narrow-lane,    -   Iono-residual,    -   Minimum error

Different linear combinations have particular characteristics that makethem useful for some aspects of ambiguity resolution or positioning.

For the full ambiguity vector to be fixed, the G matrix would simply bethe identity matrix of the proper dimension.

$\begin{matrix}{{G_{Full} = \begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}}{and}} & (31) \\{{G_{Full} \cdot \overset{\Cap}{a}} = \overset{\Cap}{a}} & (32)\end{matrix}$

Some embodiments of the invention employ the partial fixing techniquesdescribed in WO 2009/058213 A2 for generation of possible ambiguitysubsets to be used as candidate sets in forming a weighted average.Thus, the ambiguity subsets can be created of some embodiments of theinvention by removing from the GNSS data set the observations of somesatellites, of a complete GNSS, of individual observations(frequencies), etc., and/or by transforming the ambiguities into integerlinear combinations, for example the widelane carrier phase combination.

A partial fixing scheme PF in accordance with some embodiments of theinvention comprises generating multiple nominee partial ambiguity setscharacterized by their G matrix:PF⊂{G_(nom)εR^(n) ^(nom) ^(×n):rankG_(nom)=n_(nom)}  (33)

To avoid confusion with the candidate ambiguities, the term “nomineepartial fixing subset” is used to designate a possible subset ofambiguities to be fixed.

In accordance with some embodiments of the invention, only the bestcandidate is computed for each ambiguity nominee subset. In accordancewith some embodiments of the invention, the quality criteria forselecting the best subset are based on the norm (chi-square) of thatbest candidate with respect to the (partial) variance-covariance matrix.

In accordance with some embodiments of the invention, integer leastsquares is applied to the part to be fixed of the float solution vectortogether with the corresponding variance-covariance matrix to get thebest candidate solution.

For G_(nom)εR^(n) ^(nom) ^(×n) the norm of the best candidate solutionfor this nominee partial fixing subset is computed in the usual way:

$\begin{matrix}{\chi_{1,G_{nom}}^{2} = {\min\limits_{z \in Z^{n_{nom}}}{\left( {z - {G_{nom} \cdot \overset{\Cap}{a}}} \right)^{T}\left( {G_{nom} \cdot Q_{\overset{\_}{a}} \cdot G_{nom}^{T}} \right)^{- 1}\left( {z - {G_{nom} \cdot \overset{\Cap}{a}}} \right)}}} & (34)\end{matrix}$

The associated best candidate ambiguity vector would be:

$\begin{matrix}{{\overset{\Cup}{a}}_{1,G_{nom}} = {\underset{z \in Z^{n_{nom}}}{argmin}{\left( {z - {G_{nom} \cdot \overset{\Cap}{a}}} \right)^{T}\left( {G_{nom} \cdot Q_{\overset{\_}{a}} \cdot G_{nom}^{T}} \right)^{- 1}\left( {z - {G_{nom} \cdot \overset{\Cap}{a}}} \right)}}} & (35)\end{matrix}$Partial Ambiguity Solution Technique

For a priori noise models perfectly matching the data the expectationvalue for the norm χ_(1,nom) ² is equal to the number of ambiguitiesn_(nom). The quotient of the actual norm divided by the number ofambiguities in any ambiguity set thus provides a quality measure thatcan be exploited:

$\begin{matrix}{C_{G_{nom}} = \frac{\chi_{1,G_{nom}}^{2}}{n_{G_{nom}}}} & (36)\end{matrix}$

In accordance with some embodiments of the invention, the ambiguitysubset with the smallest (hence best) quality measure is selected as thesubset to be used for the final ambiguity resolution step.

$\begin{matrix}{G_{best} = {\underset{G_{nom} \in {PF}}{argmin}C_{G_{nom}}}} & (37)\end{matrix}$

In accordance with some embodiments of the invention, a check is made todetermine whether the improvement is significant. This includes testingthe improvement factor of the quality criterion

$\frac{G_{Full}}{G_{best}}$against a minimum factor. In accordance with some embodiments of theinvention it also includes comparison with a minimal criterion for thefull solution to decide if it is worthwhile to try partial fixing atall.

Sometimes, an insufficient over-determination in the floating solutioncan result in a very good fit of the best solution even for badsolutions. To avoid this, the quality criterion is limited in accordancewith some embodiments of the invention to a minimum (of, for example,1.0).

$\begin{matrix}{C_{G_{nom}}^{\prime} = {\min\left( {1,\frac{\chi_{1,G_{nom}}^{2}}{n_{G_{nom}}}} \right)}} & (38)\end{matrix}$

Once the best partial fixing subset has been determined, the ambiguitydetermination process continues using the partial solution as withoutpartial fixing.

For “traditional” ambiguity resolution (fixing of integer ambiguitiesrather than forming a weighted average of integer ambiguity candidatesets as in embodiments of the present invention), the Integer LeastSquares and validation are performed using the transformed ambiguityvector and covariance matrix. The final additional model parameters(e.g. position) are computed for the best partial solution {hacek over(a)}_(best) using the formulas given above.{hacek over (b)}={circumflex over (b)}−(G·Q_(â{circumflex over (b)}))^(T)·(G·Q _(â) ·G ^(T))⁻¹·(G·â−{hacek over(a)} _(best))  (39))andQ _({hacek over (b)}) =Q _({circumflex over (b)})−(G·Q_(â{circumflex over (b)}))^(T)·(G·Q _({hacek over (a)}) ·G ^(T))⁻¹·(G·Q_(â{circumflex over (b)}))  (40)

For ambiguity estimation in which a weighted average of integerambiguity candidate sets is formed in accordance with embodiments of thepresent invention, the transformed ambiguity vector and covariancematrix are used as well. Computing receiver position using the weightedaverage of the integer ambiguity candidate sets is carried out inaccordance with some embodiments of the present invention using amodified position gain matrix(G·Q_(â{circumflex over (b)}))^(T)·(G·Q_(â)·G^(T))⁻¹  (41)Partial Reset Technique

In accordance with some embodiments of the invention, the best normcomputation is used to detect and solve problems in the floatingsolution. If the best quality measure

$C_{G_{best}} = \frac{\chi_{1,G_{best}}^{2}}{n_{G_{best}}}$is significantly worse than the expectation value 1, a problem in thefloating solution is taken into consideration. This is based on the factthat if the best solution is already worse than statistically expected,the correct solution can only be as bad or worse. In accordance withsome embodiments of the invention the threshold value of the qualitymeasure for which is problem is declared (the “problem threshold”) is10.

In accordance with some embodiments of the invention, to identify asubset of ambiguities causing the problem, again the quality measure ofthe best set is inspected. If the improvement factor

$\frac{G_{Full}}{G_{best}}$is significantly large (for example by a factor of 4), the ambiguityestimates not included in that subset are reset in the floatingsolution. In accordance with some embodiments of the invention, this isimplemented by a large noise input for the associated ambiguity statesin the float solution. This is performed for all subfilters inembodiments of the invention in which the filter used to estimated floatambiguities is implemented as a factorized carrier-ambiguity resolutionfilter, e.g., as described in U.S. Pat. No. 7,432,853 B2.

In accordance with some embodiments of the invention, nuisance states ofthe state vector such as multipath and/or ionosphere are also reset forthe affected ambiguities/satellites.

In accordance with some embodiments of the invention, a complete resetof the float solution is performed if an identification of a singlesubset cannot be done, for example when the best improvement factor isnot large enough.

In one embodiment, the filter used to compute the float solutionestimates float ambiguities for only some of the frequencies that areobserved (or, more generally, the frequencies for which observations areobtained) and that can be tracked. In other words, the weighted averageis formed based on at least some candidate sets formed on the basis ofsome of the observed frequencies (or, more generally, the frequencieswhich observations are obtained).

In some embodiments, the iFlex solution can be computed if at least 5satellites are tracked and corresponding ambiguities are available. Ithas been recognized however that, if more than ten ambiguities are usedfor the weighted average of the candidate sets, the weighted average andthus reliability of iFlex solution is not significantly improved.

Accordingly, instead of forming the weighted average on the basis of allambiguities available, prior to forming the candidate sets, a subgroupof ambiguities may be selected, and only the selected group ofambiguities is used for forming the candidate sets that are taken intoaccount for forming the weighted average. Processing capacity can besaved (weight computation etc), while not significantly reducingreliability of the iFlex solution.

Selecting a subset of float ambiguities of the state vector can be donein various ways. In an embodiment shown in FIG. 21, the filter isoperated to estimate from the observations a complete set of floatambiguity values and subsequently the subset of float ambiguities isselected from among the complete set of float ambiguity values. In anembodiment shown in FIG. 22, the filter is operated to estimate from theobservations a selected partial set of float ambiguity values comprisingfewer than the full set of float ambiguity values, wherein the partialset of float ambiguity values comprises the subset of float ambiguitiesof the state vector.

In one embodiment, a method is provided to estimate parameters derivedfrom GNSS signals useful to determine a position. The method includesobtaining observations of a GNSS signal from each of a plurality of GNSSsatellites; feeding the observations to a filter having a state vectorat least comprising a float ambiguity for each received frequency of theGNSS signals, each float ambiguity constituting a real number estimateassociated with an integer number of wavelengths of the GNSS signalbetween a receiver of the GNSS signal and the GNSS satellite from whichit is received, and the filter being for estimating a float value foreach float ambiguity of the state vector; selecting a subset of floatambiguities of the state vector; assigning integer values to theestimated float values of the float ambiguities of the subset to definea plurality of integer ambiguity candidate sets; determining a qualitymeasure for each of the candidate sets; and obtaining a weighted averageof the candidate sets.

The advantages of this aspect of the invention notably include thedecrease in computation burden. This is because fewer carrier phaseambiguities are taken into account in the filter.

In one embodiment, as illustrated in FIG. 21, a method 4100 is provided.The method estimates parameters derived from GNSS signals useful todetermine the position. The considerations regarding theZ-transformation, as mentioned above with respect to other embodiments,also apply to the present embodiments. The method includes a step 4120of obtaining observations of a GNSS signal from each of a plurality ofGNSS satellites. This step 4120 corresponds to the step 120 of FIG. 5 a,the step 1120 of FIG. 11 a and the step 2120 of FIG. 11, and thedescription applying to these steps also applies to step 4120.

The method also includes a step 4140 of feeding the observations into afilter having a state vector at least comprising a float ambiguity foreach received frequency of the GNSS signals. Each float ambiguityconstitutes a real number estimate associated with an integer number ofwavelengths of the GNSS signal between a receiver of the GNSS signal andthe GNSS satellite from which it is received. The filter estimates afloat value for each float ambiguity of the state vector.

The method further includes a step 4150 of selecting a subset of theambiguities of the float solution, but not all float ambiguities of thefloat solution. In one embodiment, at least five float ambiguities areselected. In one embodiment, the minimum required number of floatambiguities are selected. The actual minimum number may depend onfactors such static or kinematic calculations, three-dimensional (3D) orknown height positioning, and so on. For instance, a minimum number ofsix float ambiguities may be determined to be required, at the expenseof a lesser yield, or a minimum number of four float ambiguities at theexpense of accepting that, for the lowest number of float ambiguities(e.g. four), the results may not be completely satisfactory.

Then, a step 4160 of assigning integer values to the estimated floatvalues of the float ambiguities of the subset is provided to define theplurality of integer ambiguity candidate sets.

A quality measure for each of the candidate sets is then determined.Finally, a weighted average of the candidate sets is obtained in step4200 to form the iFlex solution, as described with reference 200 of FIG.5 a.

In one embodiment, illustrated in FIG. 22, of the step 4150, whenfeeding 4140 the observations into a filter, a subset of the ambiguitiesare selected 4150 to form the state vector and therefore the floatsolution.

These embodiments, namely the embodiments illustrated with reference toFIGS. 21 and 22, enable to reduce the computation processing burdenwithout significantly reducing the position precision.

In one embodiment, the step of selecting 4150 includes selecting, asfloat ambiguities of the subset, the float ambiguities of thefrequencies which have been tracked continuously for the longest periodof time. Other criteria to select float ambiguities to be considered inthe filter may be used, with in mind the goal of taking into account theobservations being the most reliable ones.

In one embodiment of the method, the selecting includes selecting, asfloat ambiguities of the subset, the float ambiguities of frequenciestracked continuously for the longest period of time.

In one embodiment of the method, the selecting includes selecting atleast five float ambiguities but fewer than all available floatambiguities. In other words, not all available float ambiguities areselected.

One aspect of the invention further includes an apparatus to estimateparameters derived from global navigational satellite system (GNSS)signals useful to determine a position, including a receiver adapted toobtain observations of a GNSS signal from each of a plurality of GNSSsatellites; a filter having a state vector at least comprising a floatambiguity for each received frequency of the GNSS signals, each floatambiguity constituting a real number estimate associated with an integernumber of wavelengths of the GNSS signal between a receiver of the GNSSsignal and the GNSS satellite from which it is received, and the filterbeing for estimating a float value for each float ambiguity of the statevector; a processing element adapted, capable or configured to

-   -   select a subset of float ambiguities of the state vector;    -   assign integer values to the estimated float values of the float        ambiguities of the subset to define a plurality of integer        ambiguity candidate sets;    -   determine a quality measure for each of the candidate sets; and    -   obtain a weighted average of the candidate sets.

The apparatus can be configured to select a subset of float ambiguitiesin various ways. In an embodiment shown in FIG. 21, the filterconfigured to estimate from the observations a complete set of floatambiguity values and the apparatus is configured to subsequently selectthe subset of float ambiguities from among the complete set of floatambiguity values. In an embodiment shown in FIG. 22, the filter isconfigured to estimate from the observations a selected partial set offloat ambiguity values comprising fewer than the full set of floatambiguity values, wherein the partial set of float ambiguity valuescomprises the subset of float ambiguities of the state vector.

In one embodiment of the apparatus, the processing element is adapted,capable or configured to select, as float ambiguities of the subset, thefloat ambiguities of frequencies tracked continuously for the longestperiod of time.

In one embodiment of the apparatus, the step of selecting a subset offloat ambiguities of the state vector includes selecting at least fivefloat ambiguities of the state vector.

As explained in more details in section 6 entitled “[6. Combination ofaspects and embodiments, and further considerations applicable to theabove]”, the receiver, the filter and the processing element of theabove-described apparatuses may be separate from each other. As alsoexplained in more details in section 6, the invention also relates to acomputer program, to a computer program medium, to a computer programproduct and to a firmware update containing code instructions forcarrying out any one of the above-described methods.

[6. Combination of Aspects and Embodiments, and Further ConsiderationsApplicable to the Above]

FIG. 23 is a schematic block diagram of a typical integrated GNSSreceiver system 2300 with GNSS antenna 2305 and communications antenna2310. Receiver system 2300 can serve as a rover or base station orreference station. Receiver system 2300 includes a GNSS receiver 2315, acomputer system 2320 and one or more communications links 2325. Computersystem 2320 includes one or more processors 2330, one or more datastorage elements 2335, program code 2340 for controlling theprocessor(s) 2330, and user input/output devices 2345 which may includeone or more output devices 2350 such as a display or speaker or printerand one or more devices 2355 for receiving user input such as a keyboardor touch pad or mouse or microphone. The processor(s) 2330 are adaptedby the program code 2340 to carry out processing functions describedherein, such as 1. candidate selection; 2. scaling of quality measure;3. indication of convergence of weighted average solution; 4. keepinglegacy observations after interruption of tracking; and/or 5. ambiguityselection.

FIG. 24 schematically illustrates a network positioning scenario 2400using a GNSS rover 2405 (such as integrated receiver system 2300) inaccordance with some embodiments of the invention. Rover 2405 receivesGNSS signals from satellites 2410 and 2415 of a first GNSS 2418,receives GNSS signals from satellites 2420 and 2425 of a second GNSS2428, and/or receives GNSS signals from satellites 2430 and 2435 of athird GNSS 2438. Rover 2405 may receive GNSS signals from satellites offurther GNSS as available. Similarly, GNSS reference stations 2440, 2445(and possibly others not shown) receive GNSS signals from some or all ofthe same satellites. A network processor 2450 collects the data from thereference receivers, prepares correction data and transmits thecorrection data to rover 2405 via a communications link 2455. Inaccordance with some embodiments, network processor 2450 is configuredas computer system 2320 having, for example, one or more processors2330, one or more data storage elements 2335, program code 2340 forcontrolling the processor(s) 2330, and user input/output devices 2345which may include one or more output devices 2350 such as a display orspeaker or printer and one or more devices 2355 for receiving user inputsuch as a keyboard or touch pad or mouse or microphone. In accordancewith some embodiments, network processor 2450 forms a part of anintegrated receiver system as illustrated in FIG. 23. In accordance withsome embodiments, network processor 2450 is a computer system which isseparate from the GNSS receivers. The processor(s) of network processor2450 are adapted by the program code 2340 to carry out processingfunctions described herein, such as 1. candidate selection; 2. scalingof quality measure; 3. indication of convergence of weighted averagesolution; 4. keeping legacy observations after interruption of tracking;and/or 5. ambiguity selection.

FIG. 25 schematically illustrates a real-time-kinematic positioningscenario 2500 using a GNSS rover 2505 capable of receiving GNSS signalsfrom GNSS satellites in view. For example, rover 2505 receives GNSSsignals from satellites 2510 and 2515 of a first GNSS 2518, receivesGNSS signals from satellites 2520 and 2525 of a second GNSS 2528, and/orreceives GNSS signals from satellites 2530 and 2535 of a third GNSS2538. Rover 2505 may receive GNSS signals from satellites of furtherGNSS as available. Similarly, GNSS base station 2540 receives GNSSsignals from some or all of the same satellites. GNSS base station 2540prepares correction data and transmits the correction data to rover 2505via a communications antenna 2555 or other suitable communications link.In accordance with some embodiments, either or both of GNSS rover 2505and GNSS base station 2540 is/are configured as integrated receiversystem 2300 each having, for example, one or more processors 2330, oneor more data storage elements 2335, program code 2340 for controllingthe processor(s) 2330, and user input/output devices 2345 which mayinclude one or more output devices 2350 such as a display or speaker orprinter and one or more devices 2355 for receiving user input such as akeyboard or touch pad or mouse or microphone. In accordance with someembodiments, network processor 2450 forms a part of an integratedreceiver system as illustrated in FIG. 23. Either or both of GNSS rover2505 and GNSS base station 2540 is/are are adapted by the program code2340 to carry out processing functions described herein, such as 1.candidate selection; 2. scaling of quality measure; 3. indication ofconvergence of weighted average solution; 4. keeping legacy observationsafter interruption of tracking; and/or 5. ambiguity selection.

Any plurality of the above described aspects of the invention includingthose described in sections “1. Candidate set selection”, “2. Scaling ofquality measure”, “3. Indication of convergence of weighted averagesolution”, “4. Keeping legacy observations after interruption oftracking” and “5. Ambiguity selection” respectively, (i.e. two, three,four or five of these aspects), may be combined to form further aspectsand embodiments, with the aim of providing additional benefits notablyin terms of computation speed, precision estimation and systemusability.

Any of the above-described apparatuses and their embodiments may beintegrated into a rover, a receiver or a network station, and/or theprocessing methods described can be carried out in a processor which isseparate from and even remote from the receivers used to collect theobservations (e.g., observation data collected by one or more receiverscan be retrieved from storage for post-processing, or observations frommultiple network reference stations can be transferred to a networkprocessor for near-real-time processing to generate a correction datastream and/or virtual-reference-station messages which can betransmitted to one or more rovers). Therefore, the invention alsorelates to a rover, a receiver or a network station including any one ofthe above apparatuses.

In one embodiment, the receiver of the apparatus of any one of theabove-described embodiments is separate from the filter and theprocessing element. Post-processing and network processing of theobservations may notably be performed. That is, the constituent elementsof the apparatus for processing of observations does not itself requirea receiver. The receiver may be separate from and even owned/operated bya different entity than the entity which is performing the processing.For post-processing, the observations may be retrieved from a set ofdata which was previously collected and stored, and processed withreference-station data which was previously collected and stored; theprocessing is conducted for example in an office computer long after thedata collection and is thus not real-time. For network processing,multiple reference-station receivers collect observations of the signalsfrom multiple satellites, and this data is supplied to a networkprocessor which may for example generate a correction data stream orwhich may for example generate a “virtual reference station” correctionwhich is supplied to a rover so that the rover can perform differentialprocessing. The data provided to the rover may be ambiguities determinedin the network processor, which the rover may use to speed its positionsolution, or may be in the form of corrections which the rover appliesto improve its position solution. The network is typically operated as aservice to rover operators, while the network operator is typically adifferent entity than the rover operator. This applies to each of theabove-described apparatuses and claims.

Any of the above-described methods and their embodiments may beimplemented by means of a computer program. The computer program may beloaded on an apparatus, a rover, a receiver or a network station asdescribed above. Therefore, the invention also relates to a computerprogram, which, when carried out on an apparatus, a rover, a receiver ora network station as described above, carries out any one of the aboveabove-described methods and their embodiments.

The invention also relates to a computer-readable medium or acomputer-program product including the above-mentioned computer program.The computer-readable medium or computer-program product may forinstance be a magnetic tape, an optical memory disk, a magnetic disk, amagneto-optical disk, a CD ROM, a DVD, a CD, a flash memory unit or thelike, wherein the computer program is permanently or temporarily stored.The invention also relates to a computer-readable medium (or to acomputer-program product) having computer-executable instructions forcarrying out any one of the methods of the invention. Acomputer-readable medium may comprise one of: a computer-readablephysical storage medium embodying a computer program and acomputer-readable transmission medium embodying a computer program.

The invention also relates to a firmware update adapted to be installedon receivers already in the field, i.e. a computer program which isdelivered to the field as a computer program product. This applies toeach of the above-described methods and apparatuses.

GNSS receivers may include an antenna, configured to received thesignals at the frequencies broadcasted by the satellites, processorunits, one or more accurate clocks (such as crystal oscillators), one ormore computer processing units (CPU), one or more memory units (RAM,ROM, flash memory, or the like), and a display for displaying positioninformation to a user.

Where the terms “receiver”, “filter” and “processing element” are usedherein as units of an apparatus, no restriction is made regarding howdistributed the constituent parts of a unit may be. That is, theconstituent parts of a unit may be distributed in different software orhardware components or devices for bringing about the intended function.Furthermore, the units may be gathered together for performing theirfunctions by means of a combined, single unit. For instance, thereceiver, the filter and the processing element may be combined to forma single unit, to perform the combined functionalities of the units.

The above-mentioned units may be implemented using hardware, software, acombination of hardware and software, pre-programmed ASICs(application-specific integrated circuit), etc. A unit may include acomputer processing unit (CPU), a storage unit, input/output (I/O)units, network connection units, etc.

Although the present invention has been described on the basis ofdetailed examples, the detailed examples only serve to provide theskilled person with a better understanding, and are not intended tolimit the scope of the invention. The scope of the invention is muchrather defined by the appended claims.

Additional features, and combinations of features, of embodiments of theinvention are as follows:

(1^(st) Aspect: Candidate Set Selection)

-   1. Method to estimate parameters derived from global navigational    satellite system (GNSS) signals useful to determine a position,    including obtaining observations of a GNSS signal from each of a    plurality of GNSS satellites;    -   feeding the observations to a filter having a state vector at        least comprising a float ambiguity for each received frequency        of the GNSS signals, each float ambiguity constituting a real        number estimate associated with an integer number of wavelengths        of the GNSS signal between a receiver of the GNSS signal and the        GNSS satellite from which it is received, and the filter being        for estimating a float value for each float ambiguity of the        state vector;    -   assigning integer values to at least a subgroup of the estimated        float values to define a plurality of integer ambiguity        candidate sets;    -   selecting a first number of candidate sets having a quality        measure better than a first threshold, wherein the first        threshold is determined based on a reference quality measure of        a reference candidate set; and    -   obtaining a weighted average of the selected candidate sets,        each candidate set weighted in the weighted average based on its        quality measure.-   2. Method of 1, including using the weighted average to estimate a    position of the receiver of the GNSS signals.-   3. Method of at least one of 1 and 2, wherein the reference    candidate set is the candidate set having the best quality measure.-   4. Method of at least one of 1 to 3, wherein the quality measure of    the candidate sets is constituted by a residual error norm value,    the residual error norm value of a candidate set being a measure for    a statistical distance of the candidate set to the state vector    having the float ambiguities.-   5. Method of at least one of 1 to 4, wherein the first threshold is    determined as at least one of a fraction of, a multiple of, and a    distance to the reference quality measure.-   6. Method of at least one of 1 to 5, including, if the first number    of selected candidate sets is smaller than a second threshold,    selecting, for forming the weighted average, a second number of    further candidate sets on the basis of the quality measures of the    candidate sets in decreasing order starting with the non-selected    candidate set having the best quality measure, the second number    being constituted by the difference between the first number of    selected candidate sets and a second threshold defining a minimum    number of candidate sets to be included in the weighted average.-   7. Method of at least one of 1 to 6, including, if the first number    of selected candidate sets is larger than a third threshold,    excluding a third number of selected candidate sets from forming the    weighted average in decreasing order starting with the selected    candidate set having the worst quality measure, wherein the third    number is constituted by the difference between the first number of    selected candidate sets and a third threshold defining a maximum    number of candidate sets to be included in the weighted average.-   8. Method of at least one of 1 to 7, wherein selecting a first    number of candidate sets includes:    -   determining the best quality measure of the candidate sets;    -   determining an expectation value of the candidate set having the        best quality measure;    -   determining an error measure as a ratio of the best quality        measure to the expectation value; adapting the quality measures        of the candidate sets as a function of the error measure; and    -   performing the selection of the first number of candidate sets        based on the adapted quality measures.-   9. Method of at least one of 1 to 8, wherein obtaining the weighted    average includes:    -   determining the best quality measure of the candidate sets;    -   determining an expectation value of the candidate set having the        best quality measure;    -   determining an error measure as a ratio of the best quality        measure to the expectation value; adapting the quality measures        of the candidate sets as a function of the error measure; and    -   forming the weighted average based on the adapted quality        measure.-   10. Method of at least one of 8 and 9, including adapting the    quality measures of the candidate sets as a function of the error    measure by adjusting a variance-covariance matrix of the filter.-   11. Method of at least one of 8 to 10, including adjusting a    variance-covariance matrix of the filter using the error measure, if    the error measure is in a predetermined range.-   12. Method of at least one of 8 to 11, including adjusting the    variance-covariance matrix of the float solution using the error    measure, if the error measure is larger than one.-   13. Method of at least one of 1 to 12, wherein defining the integer    candidate sets includes:    -   selecting a subset of float ambiguities of the state vector to        form a subgroup of the float ambiguities of the state vector        used to define the plurality of integer ambiguity candidate        sets; and    -   assigning integer values to the estimated float values of the        float ambiguities of the subset to define a plurality of integer        ambiguity candidate sets.-   14. Method of 13, wherein the selecting includes selecting, as float    ambiguities of the subset, the float ambiguities of frequencies    tracked continuously for the longest period of time.-   15. Method of at least one of 1 to 14, including    -   estimating by the filter a float value for each float ambiguity        of the state vector and covariance values associated with the        state vector;    -   determining a formal precision value based on covariance values        of the filter, the formal precision value being a measure for an        achievable precision;    -   determining an achieved precision value of the weighted average;    -   comparing the achieved precision value with the formal precision        value to obtain a convergence value; and    -   indicating a convergence of the determination of the state        vector based on the convergence value.-   16. Method of 15, wherein the convergence value is obtained as a    ratio of the achieved precision value to the formal precision value.-   17. Method of at least one of 15 and 16, including    -   determining an instance in time when the convergence value of        the position is better than a convergence threshold; and    -   indicating a convergence of the determination of the state        vector at and after the determined instance in time.-   18. Method of at least one of 15 to 17, including    -   estimating an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicating a convergence of the determination of the state        vector, if the achieved precision of the position is better than        an inclusion threshold.-   19. Method of at least one of 15 to 18, including    -   estimating an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicating a non-convergence of the determination of the state        vector, if the achieved precision of the position is worse than        an exclusion threshold.-   20. Method of at least one 1 to 19, including:    -   obtaining observations of the at least one frequency of the GNSS        signals from the plurality of GNSS satellites to obtain        observations at a plurality of instances in time;    -   updating the float ambiguities of the state vector over time on        the basis of the observations;    -   determining that an interruption in tracking of at least one        signal of a satellite occurred; and    -   maintaining the float ambiguity of the state vector for the at        least one signal for which an interruption in tracking occurred        at the value before the interruption in tracking occurred.-   21. Method of 20, wherein it is determined that an interruption in    tracking of at least one signal of a satellite occurred if an    observation for the at least one signal is not available for at    least one of the instances in time.-   22. Method of at least one of 20 and 21, wherein it is determined    that an interruption in tracking of at least one signal of a    satellite occurred, if a cycle slip occurred.-   23. Method of at least one of 20 to 22 including, if, after an    interruption in tracking of a signal, tracking of the signal    resumes, maintaining the float ambiguity of the state vector for the    signal for which an interruption in tracking occurred at the value    before the interruption in tracking occurred as a first float    ambiguity and introducing into the state vector a second float    ambiguity for the signal after resuming tracking.-   24. Apparatus to estimate parameters derived from global    navigational satellite system (GNSS) signals useful to determine a    position, including    -   a receiver of a GNSS signal from each of a plurality of GNSS        satellites;    -   a filter having a state vector at least comprising a float        ambiguity for each received frequency of the GNSS signals, each        float ambiguity constituting a real number estimate associated        with an integer number of wavelengths of the GNSS signal between        a receiver of the GNSS signal and the GNSS satellite from which        it is received, and the filter being for estimating a float        value for each float ambiguity of the state vector;    -   a processing element adapted to        -   assign integer values to at least a subgroup of the            estimated float values to define a plurality of integer            ambiguity candidate sets;        -   select a first number of candidate sets having a quality            measure better than a first threshold, wherein the first            threshold is determined based on a reference quality measure            of a reference candidate set; and        -   obtain a weighted average of the selected candidate sets,            each candidate set being weighted in the weighted average            based on its quality measure.-   25. Apparatus of 24, wherein the processing element is adapted to    use the weighted average to estimate a position of the receiver of    the GNSS signals.-   26. Apparatus of at least one of 24 and 25, wherein the reference    candidate set is the candidate set having the best quality measure.-   27. Apparatus of at least one of 24 to 26, wherein the quality    measure of the candidate sets is constituted by a residual error    norm value, the residual error norm value of a candidate set being a    measure for a statistical distance of the candidate set to the state    vector having the float ambiguities.-   28. Apparatus of at least one of 24 to 27, wherein the processing    element is adapted to determine the first threshold as at least one    of a fraction of, a multiple of, and a distance to the reference    quality measure.-   29. Apparatus of at least one of 24 to 28, wherein the processing    element is adapted to, if the first number of selected candidate    sets is smaller than a second threshold, selecting, from the    weighted average, a second number of further candidate sets on the    basis of the quality measures of the candidate sets in decreasing    order starting with the non-selected candidate set having the best    quality measure, the second number being constituted by the    difference between the first number of selected candidate sets and a    second threshold defining a minimum number of candidate sets to be    included in the weighted average.-   30. Apparatus of at least one of 24 to 29, wherein the processing    element is adapted to, if the first number of selected candidate    sets is larger than a third threshold, exclude a third number of    selected candidate sets from forming the weighted average in    decreasing order starting with the selected candidate set having the    worst quality measure, wherein the third number is constituted by    the difference between the first number of selected candidate sets    and a third threshold defining a maximum number of candidate sets to    be included in the weighted average.-   31. Apparatus of at least one of 24 to 30, wherein the processing    element is adapted to, in order to select a first number of    candidate sets:    -   determine the best quality measure of the candidate sets;    -   determine an expectation value of the candidate set having the        best quality measure;    -   determine an error measure as a ratio of the best quality        measure to the expectation value;    -   adapt the quality measures of the candidate sets as a function        of the error measure; and    -   perform the selection of the first number of candidate sets        based on the adapted quality measures.-   32. Apparatus of at least one of 24 to 31, wherein the processing    element is adapted to, in order to obtain the weighted average:    -   determine the best quality measure of the candidate sets;    -   determine an expectation value of the candidate set having the        best quality measure;    -   determine an error measure as a ratio of the best quality        measure to the expectation value;    -   adapt the quality measures of the candidate sets as a function        of the error measure; and    -   form the weighted average based on the adapted quality measure.-   33. Apparatus of at least one of 31 and 32, wherein the processing    element is adapted to adapt the quality measures of the candidate    sets as a function of the error measure by adjusting the    variance-covariance matrix of the float solution.-   34. Apparatus of at least one of 31 to 33, wherein the processing    element is adapted to adjust the variance-covariance matrix of the    float solution using the error measure, if the error measure is in a    predetermined range.-   35. Apparatus of at least one of 31 to 34, wherein the processing    element is adapted to adjust the variance-covariance matrix of the    float solution using the error measure, if the error measure is    larger than one.-   36. Apparatus of at least one of 24 to 35, wherein the processing    element is adapted to, in order to define the integer candidate    sets:    -   select a subset of float ambiguities of the state vector to form        a subgroup of the float ambiguities of the state vector used to        define the plurality of integer ambiguity candidate sets; and    -   assign integer values to the estimated float values of the float        ambiguities of the subset to define a plurality of integer        ambiguity candidate sets.-   37. Apparatus of 36, wherein the processing element is adapted to    select, as float ambiguities of the subset, the float ambiguities of    frequencies tracked continuously for the longest period of time.-   38. Apparatus of at least one of 24 to 37, wherein    -   the filter is adapted to        -   estimate a float value for each float ambiguity of the state            vector and covariance values associated with the state            vector; and    -   the processing element is adapted to        -   determine a formal precision value based on covariance            values of the filter, the formal precision value being a            measure for an achievable precision;        -   determine an achieved precision value of the weighted            average;        -   compare the achieved precision value with the formal            precision value to obtain a convergence value; and        -   indicate a convergence of the determination of the state            vector based on the convergence value.-   39. Apparatus of 38, wherein the processing element is adapted to    obtain the convergence value as a ratio of the achieved precision    value to the formal precision value.-   40. Apparatus of at least one of 38 and 39, wherein the processing    element is adapted to    -   determine an instance in time when the achieved precision of the        position is better than a convergence threshold; and    -   indicate a convergence of the determination of the state vector        at and after the determined instance in time.-   41. Apparatus of at least one of 38 to 40, wherein the processing    element is adapted to    -   estimate an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicate a convergence of the determination of the state vector,        if the achieved precision of the position is better than an        inclusion threshold.-   42. Apparatus of at least one of 38 to 41, wherein the processing    element is adapted to    -   estimate an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicate a non-convergence of the determination of the state        vector, if the achieved precision of the position is worse than        an exclusion threshold.-   43. Apparatus of at least one of 24 to 42, adapted to:    -   obtain observations of the at least one frequency of the GNSS        signals from the plurality of GNSS satellites to obtain        observations at a plurality of instances in time;    -   update the float ambiguities of the state vector over time on        the basis of the observations; and    -   determine that an interruption in tracking of at least one        signal of a satellite occurred; and    -   maintain the float ambiguity of the state vector for the at        least one signal for which an interruption in tracking occurred        at the value before the interruption in tracking occurred.-   44. Apparatus of 43, adapted to determine that an interruption in    tracking of at least one signal of a satellite occurred if an    observation for the at least one signal is not available for at    least one of the instances in time.-   45. Apparatus of at least one of 43 and 44, adapted to determine    that an interruption in tracking of at least one signal of a    satellite occurred, if a cycle slip occurred.-   46. Apparatus of at least one of 43 to 45, adapted to, if, after an    interruption in tracking of a signal, tracking of the signal    resumes, maintain the float ambiguity of the state vector for the    signal for which an interruption in tracking occurred at the value    before the interruption in tracking occurred as a first float    ambiguity and introducing into the state vector a second float    ambiguity for the signal after resuming tracking.-   47. Rover including an apparatus according to any one of 24 to 46.-   48. Network station including an apparatus according to any one of    24 to 46.-   49. Computer program comprising instructions configured, when    executed on a computer processing unit, to carry out a method    according to any one of 1 to 23.-   50. Computer-readable medium comprising a computer program according    to 49.

(2^(nd) Aspect: Scaling of Quality Measure)

-   1. Method to estimate parameters derived from global navigational    satellite system (GNSS) signals useful to determine a position,    -   obtaining observations of a GNSS signal from each of a plurality        of GNSS satellites;    -   feeding the observations to a filter having a state vector at        least comprising a float ambiguity for each received frequency        of the GNSS signals, each float ambiguity constituting a real        number estimate associated with an integer number of wavelengths        of the GNSS signal between a receiver of the GNSS signal and the        GNSS satellite from which it is received, and the filter being        for estimating a float value for each float ambiguity of the        state vector;    -   assigning integer values to at least a subgroup of the estimated        float values to define a plurality of integer ambiguity        candidate sets;    -   determining a quality measure for each of the candidate sets;    -   determining the best quality measure of the candidate sets;    -   determining an expectation value of the candidate set having the        best quality measure;    -   determining an error measure as a ratio of the best quality        measure to the expectation value;    -   adapting the quality measures of the candidate sets as a        function of the error measure; and    -   obtaining a weighted average of a subgroup of the candidate sets        on the basis of the adapted quality measures, wherein at least        one of selecting the subgroup of the candidate sets and the        weighting of each candidate set in the weighted average is based        on the adapted quality measure.-   2. Method of 1, including adapting the quality measures of the    candidate sets as a function of the error measure by adjusting the    variance-covariance matrix of the float solution.-   3. Method of at least one of the preceding claims, including    adjusting a variance-covariance matrix of the filter using the error    measure, if the error measure is in a predetermined range.-   4. Method of at least one of the preceding claims, including    adjusting a variance-covariance matrix of the filter using the error    measure, if the error measure is larger than one.-   5. Method of at least one of 1 to 4, wherein obtaining the weighted    average includes:    -   selecting a first number of candidate sets having a quality        measure better than a first threshold to form the subgroup,        wherein the first threshold is determined based on a reference        quality measure a reference candidate set; and    -   obtaining the weighted average of the selected candidate sets of        the subgroup, each candidate set weighted in the weighted        average based on its quality measure.-   6. Method of at least one of 1 to 5, including using the weighted    average to estimate a position of the receiver of the GNSS signals.-   7. Method of at least one of 5 and 6, wherein the reference    candidate set is the candidate set having the best quality measure.-   8. Method of at least one of 5 to 7, wherein the quality measure of    the candidate sets is constituted by the residual error norm value,    the residual error norm value of a candidate set being a measure for    a statistical distance of the candidate set to the state vector    having the float ambiguities.-   9. Method of at least one of 5 to 8, wherein the first threshold is    determined as at least one of a fraction of, a multiple of, and a    distance to the reference quality measure.-   10. Method of at least one of 5 to 9, including, if the first number    of selected candidate sets is smaller than a second threshold,    selecting, for forming the weighted average, a second number of    further candidate sets on the basis of the quality measures of the    candidate sets in decreasing order starting with the non-selected    candidate set having the best quality measure, the second number    being constituted by the difference between the first number of    selected candidate sets and a second threshold defining a minimum    number of candidate sets to be included in the weighted average.-   11. Method of at least one of 5 to 10, including, if the first    number of selected candidate sets is larger than a third threshold,    excluding a third number of selected candidate sets from forming the    weighted average in decreasing order starting with the selected    candidate set having the worst quality measure, wherein the third    number is constituted by the difference between the first number of    selected candidate sets and a third threshold defining a maximum    number of candidate sets to be included in the weighted average.-   12. Method of at least one of Ito 11, wherein defining the integer    candidate sets includes:    -   selecting a subset of float ambiguities of the state vector to        form the subgroup of the float ambiguities of the state vector        used to define the plurality of integer ambiguity candidate        sets; and    -   assigning integer values to the estimated float values of the        float ambiguities of the subset to define a plurality of integer        ambiguity candidate sets.-   13. Method of 12, wherein the selecting includes selecting, as float    ambiguities of the subset, the float ambiguities of frequencies    tracked continuously for the longest period of time.-   14. Method of at least one 1 to 13, including estimating by the    filter a float value for each float ambiguity of the state vector    and covariance values associated with the state vector;    -   determining a formal precision value based on covariance values        of the filter, the formal precision value being a measure for an        achievable precision;    -   determining an achieved precision value of the weighted average;    -   comparing the achieved precision value with the formal precision        value to obtain a convergence value; and    -   indicating a convergence of the determination of the state        vector based on the convergence value.-   15. Method of 14, wherein the convergence value is obtained as a    ratio of the achieved precision value to the formal precision value.-   16. Method of at least one of 14 and 15, including    -   determining an instance in time when the convergence value of        the position is better than a convergence threshold; and    -   indicating a convergence of the determination of the state        vector at and after the determined instance in time.-   17. Method of at least one of 14 to 16, including    -   estimating an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicating a convergence of the determination of the state        vector, if the achieved precision of the position is better than        an inclusion threshold.-   18. Method of at least one of 14 to 17, including    -   estimating an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicating a non-convergence of the determination of the state        vector, if the achieved precision of the position is worse an        exclusion threshold.-   19. Method of at least one of 1 to 18, including    -   obtaining observations of the at least one frequency of the GNSS        signals from the plurality of GNSS satellites to obtain        observations at a plurality of instances in time;    -   updating the float ambiguities of the state vector over time on        the basis of the observations;    -   determining that an interruption in tracking of at least one        signal of a satellite occurred; and    -   maintaining the float ambiguity of the state vector for the at        least one signal for which an interruption in tracking occurred        at the value before the interruption in tracking occurred.-   20. Method of 19, wherein it is determined that an interruption in    tracking of at least one signal of a satellite occurred if an    observation for the at least one signal is not available for at    least one of the instances in time.-   21. Method of at least one of 19 and 20, wherein it is determined    that an interruption in tracking of at least one signal of a    satellite occurred, if a cycle slip occurred.-   22. Method of at least one of 19 to 21 including, if after an    interruption in tracking of a signal tracking of the signal resumes,    maintaining the float ambiguity of the state vector for the signal    for which an interruption in tracking occurred at the value before    the interruption in tracking occurred as a first float ambiguity and    introducing into the state vector a second float ambiguity for the    signal after resuming tracking.-   23. Apparatus to estimate parameters derived from global    navigational satellite system (GNSS) signals useful to determine a    position,    -   a receiver adapted to obtain observations of a GNSS signal from        each of a plurality of GNSS satellites;    -   a filter having a state vector at least comprising a float        ambiguity for each received frequency of the GNSS signals, each        float ambiguity constituting a real number estimate associated        with an integer number of wavelengths of the GNSS signal between        a receiver of the GNSS signal and the GNSS satellite from which        it is received, and the filter being for estimating a float        value for each float ambiguity of the state vector;    -   a processing element adapted to        -   assign integer values to at least a subgroup of the            estimated float values to define a plurality of integer            ambiguity candidate sets;        -   determine a quality measure for each of the candidate sets;        -   determine the best quality measure of the candidate sets;        -   determine an expectation value of the candidate set having            the best quality measure;        -   determine an error measure as a ratio of the best quality            measure to the expectation value;        -   adapt the quality measures of the candidate sets as a            function of the error measure; and        -   obtain a weighted average of a subgroup of the candidate            sets on the basis of the adapted quality measures, wherein            at least one of selecting the subgroup of the candidate sets            and the weighting of each candidate set in the weighted            average is based on the adapted quality measure.-   24. Apparatus of 23, wherein the processing element is adapted to    adapt the quality measures of the candidate sets as a function of    the error measure by adjusting the variance-covariance matrix of the    float solution.-   25. Apparatus of at least one of 23 and 24, wherein the processing    element is adapted to adjust a variance-covariance matrix of the    filter using the error measure, if the error measure is in a    predetermined range.-   26. Apparatus of at least one of 23 to 25, wherein the processing    element is adapted to adjust a variance-covariance matrix of the    filter using the error measure, if the error measure is larger than    one.-   27. Apparatus of at least one of 23 to 26, wherein the processing    element is adapted to, in order to obtain the weighted average:    -   selecting a first number of candidate sets having a quality        measure better than a first threshold to form the subgroup,        wherein the first threshold is determined based on a reference        quality measure a reference candidate set; and    -   obtaining the weighted average of the selected candidate sets of        the subgroup, each candidate set weighted in the weighted        average based on its quality measure.-   28. Apparatus of at least one of 23 to 27, including using the    weighted average to estimate a position of the receiver of the GNSS    signals.-   29. Apparatus of at least one of 27 and 28, wherein the reference    candidate set is the candidate set having the best quality measure.-   30. Apparatus of at least one of 27 to 29, wherein the quality    measure of the candidate sets is constituted by the residual error    norm value, the residual error norm value of a candidate set being a    measure for a statistical distance of the candidate set to the state    vector having the float ambiguities.-   31. Apparatus of at least one of 27 to 30, wherein the processing    element is adapted to determine the first threshold as at least one    of a fraction of, a multiple of, and a distance to the reference    quality measure.-   32. Apparatus of at least one of 27 to 31, wherein the processing    element is adapted to, if the first number of selected candidate    sets is smaller than a second threshold, select, for forming the    weighted average, a second number of further candidate sets on the    basis of the quality measures of the candidate sets in decreasing    order starting with the non-selected candidate set having the best    quality measure, the second number being constituted by the    difference between the first number of selected candidate sets and a    second threshold defining a minimum number of candidate sets to be    included in the weighted average.-   33. Apparatus of at least one of 27 to 32, wherein the processing    element is adapted to, if the first number of selected candidate    sets is larger than a third threshold, exclude a third number of    selected candidate sets from forming the weighted average in    decreasing order starting with the selected candidate set having the    worst quality measure, wherein the third number is constituted by    the difference between the first number of selected candidate sets    and a third threshold defining a maximum number of candidate sets to    be included in the weighted average.-   34. Apparatus of at least one of 23 to 33, wherein the processing    element is adapted to, in order to define the integer candidate    sets:    -   selecting a subset of float ambiguities of the state vector to        form the subgroup of the float ambiguities of the state vector        used to define the plurality of integer ambiguity candidate        sets; and    -   assigning integer values to the estimated float values of the        float ambiguities of the subset to define a plurality of integer        ambiguity candidate sets.-   35. Apparatus of 34, wherein the processing element is adapted to    select, as float ambiguities of the subset, the float ambiguities of    frequencies tracked continuously for the longest period of time:-   36. Apparatus of at least one 23 to 35, wherein    -   the filter is adapted to        -   estimate a float value for each float ambiguity of the state            vector and covariance values associated with the state            vector; and    -   the processing element is adapted to        -   determine a formal precision value based on covariance            values of the filter, the formal precision value being a            measure for an achievable precision;        -   determine an achieved precision value of the weighted            average;        -   compare the achieved precision value with the formal            precision value to obtain a convergence value; and        -   indicate a convergence of the determination of the state            vector based on the convergence value.-   37. Apparatus of 36, wherein the processing element is adapted to    obtain the convergence value as a ratio of the achieved precision    value to the formal precision value.-   38. Apparatus of at least one of 36 and 37, wherein the processing    element is adapted to    -   determine an instance in time when the convergence value of the        position is better than a convergence threshold; and    -   indicate a convergence of the determination of the state vector        at and after the determined instance in time.-   39. Apparatus of at least one of 36 to 38, wherein the processing    element is adapted to estimate an achieved precision of a position    of the receiver determined based on the weighted ambiguities; and    -   indicate a convergence of the determination of the state vector,        if the achieved precision of the position is better than an        inclusion threshold.-   40. Apparatus of at least one of 36 to 39, wherein the processing    element is adapted to estimate an achieved precision of a position    of the receiver determined based on the weighted ambiguities; and    -   indicate a non-convergence of the determination of the state        vector, if the achieved precision of the position is worse an        exclusion threshold.-   41. Apparatus of at least one of 23 to 40, adapted to    -   obtain observations of the at least one frequency of the GNSS        signals from the plurality of GNSS satellites to obtain        observations at a plurality of instances in time;    -   update the float ambiguities of the state vector over time on        the basis of the observations;    -   determine that an interruption in tracking of at least one        signal of a satellite occurred; and    -   maintain the float ambiguity of the state vector for the at        least one signal for which an interruption in tracking occurred        at the value before the interruption in tracking occurred.-   42. Apparatus of 41, adapted to determine that an interruption in    tracking of at least one signal of a satellite occurred if an    observation for the at least one signal is not available for at    least one of the instances in time.-   43. Apparatus of at least one of 41 and 42, adapted to determine    that an interruption in tracking of at least one signal of a    satellite occurred, if a cycle slip occurred.-   44. Apparatus of at least one of 41 to 43, adapted to, if, after an    interruption in tracking of a signal, tracking of the signal    resumes, maintain the float ambiguity of the state vector for the    signal for which an interruption in tracking occurred at the value    before the interruption in tracking occurred as a first float    ambiguity and introducing into the state vector a second float    ambiguity for the signal after resuming tracking.-   45. Rover including an apparatus according to any one of 23 to 44.-   46. Network station including an apparatus according to any one of    23 to 44.-   47. Computer program comprising instructions configured, when    executed on a computer processing unit, to carry out a method    according to any one of 1 to 22.-   48. Computer-readable medium comprising a computer program according    to 47.

(3^(rd) Aspect: Indication of Convergence of Weighted Average Solution)

-   1. Method to estimate parameters derived from global navigational    satellite system (GNSS) signals useful to determine a position,    including    -   obtaining observations of a GNSS signal from each of a plurality        of GNSS satellites;    -   feeding the observations to a filter having a state vector        comprising a float ambiguity for each received frequency of the        GNSS signals, each float ambiguity constituting a real number        estimate of an integer number of wavelengths of the GNSS signal        between a receiver of the GNSS signal and the GNSS satellite        from which it is received, the filter estimating a float value        for each float ambiguity of the state vector and covariance        values associated with the state vector;    -   assigning integer values to at least a subgroup of the estimated        float values to define a plurality of integer ambiguity        candidate sets;    -   obtaining a weighted average of the candidate sets;    -   determining a formal precision value based on covariance values        of the filter, the formal precision value being a measure for an        achievable precision;    -   determining an achieved precision value of the weighted average;    -   comparing the achieved precision value with the formal precision        value to obtain a convergence value; and    -   indicating a convergence of the determination of the state        vector.-   2. Method of 1, wherein the convergence value is obtained as a ratio    of the achieved precision value to the formal precision value.-   3. Method of at least one of 1 and 2, including    -   determining an instance in time when the convergence value of        the position is better than a convergence threshold; and    -   indicating a convergence of the determination of the state        vector at and after the determined instance in time.-   4. Method of at least one of 1 to 3, including    -   estimating an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicating a convergence of the determination of the state        vector, if the achieved precision of the position is better than        an inclusion threshold-   5. Method of at least one of 1 to 4, including    -   estimating an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicating a non-convergence of the determination of the state        vector, if the achieved precision of the position is worse than        an exclusion threshold.-   6. Method of at least one of 1 to 5, wherein obtaining a weighted    average of the candidate sets includes;    -   selecting a first number of candidate sets having a quality        measure better than a first threshold, wherein the first        threshold is determined based on a reference quality measure of        a reference candidate set; and    -   forming the weighted average of the selected candidate sets by        weighing each candidate set in the weighted average based on its        quality measure.-   7. Method of 6, including using the weighted average to estimate a    position of the receiver of the GNSS signals.-   8. Method of at least one of 6 and 7, wherein the reference    candidate set is the candidate set having the best quality measure.-   9. Method of at least one of 6 to 8, wherein the quality measure of    the candidate sets is constituted by a residual error norm value,    the residual error norm value of a candidate set being a measure for    a statistical distance of the candidate set to the state vector    having the float ambiguities.-   10. Method of at least one of 6 to 9, wherein the first threshold is    determined as at least one of a fraction of, a multiple of, and a    distance to the reference quality measure.-   11. Method of at least one of 6 to 10, including, if the first    number of selected candidate sets is smaller than a second    threshold, selecting, for forming the weighted average, a second    number of further candidate sets on the basis of the quality    measures of the candidate sets in decreasing order starting with the    non-selected candidate set having the best quality measure, the    second number being constituted by the difference between the first    number of selected candidate sets and a second threshold defining a    minimum number of candidate sets to be included in the weighted    average.-   12. Method of at least one of 6 to 11, including, if the first    number of selected candidate sets is larger than a third threshold,    excluding a third number of selected candidate sets from forming the    weighted average in decreasing order starting with the selected    candidate set having the worst quality measure, wherein the third    number is constituted by the difference between the first number of    selected candidate sets and a third threshold defining a maximum    number of candidate sets to be included in the weighted average.-   13. Method of at least one of 1 to 12, wherein obtaining the    weighted average includes:    -   determining the best quality measure of the candidate sets;    -   determining an expectation value of the candidate set having the        best quality measure;    -   determining an error measure as a ratio of the best quality        measure to the expectation value;    -   adapting the quality measures of the candidate sets as a        function of the error measure;    -   performing a selection of the first number of candidate sets        based on the adapted quality measures; and    -   forming the weighted average on the basis of the quality        measures.-   14. Method of at least one of 1 to 13, wherein obtaining the    weighted average includes:    -   determining the best quality measure of the candidate sets;    -   determining an expectation value of the candidate set having the        best quality measure;    -   determining an error measure as a ratio of the best quality        measure to the expectation value;    -   adapting the quality measures of the candidate sets as a        function of the error measure; and    -   forming the weighted average based on the adapted quality        measures.-   15. Method of at least one of 13 and 14, including adapting the    quality measures of the candidate sets as a function of the error    measure by adjusting the variance-covariance matrix of the float    solution.-   16. Method of at least one of 13 to 15, including adjusting a    variance-covariance matrix of the filter using the error measure, if    the error measure is in a predetermined range.-   17. Method of at least one of 13 to 16, including adjusting a    variance-covariance matrix of the filter using the error measure, if    the error measure is larger than one.-   18. Method of at least one of 1 to 17, wherein defining the integer    candidate sets includes:    -   selecting a subset of float ambiguities of the state vector to        form the subgroup of the float ambiguities of the state vector        used to define the plurality of integer ambiguity candidate        sets; and    -   assigning integer values to the estimated float values of the        float ambiguities of the subset to define a plurality of integer        ambiguity candidate sets.-   19. Method of 18, wherein the selecting includes selecting, as float    ambiguities of the subset, the float ambiguities of frequencies    tracked continuously for the longest period of time.-   20. Method of at least one of 1 to 19, including    -   obtaining observations of the at least one frequency of the GNSS        signals from the plurality of GNSS satellites to obtain        observations at a plurality of instances in time;    -   updating the float ambiguities of the state vector over time on        the basis of the observations; determining that an interruption        in tracking of at least one signal of a satellite occurred; and    -   maintaining the float ambiguity of the state vector for the at        least one signal for which an interruption in tracking occurred        at the value before the interruption in tracking occurred.-   21. Method of 20, wherein it is determined that an interruption in    tracking of at least one signal of a satellite occurred if an    observation for the at least one signal is not available for at    least one of the instances in time.-   22. Method of at least one of 20 and 21, wherein it is determined    that an interruption in tracking of at least one signal of a    satellite occurred, if a cycle slip occurred.-   23. Method of at least one of 20 to 22 including, if, after an    interruption in tracking of a signal, tracking of the signal    resumes, maintaining the float ambiguity of the state vector for the    signal for which an interruption in tracking occurred at the value    before the interruption in tracking occurred as a first float    ambiguity and introducing into the state vector a second float    ambiguity for the signal after resuming tracking.-   24. Apparatus to estimate parameters derived from global    navigational satellite system (GNSS) signals useful to determine a    position, including    -   a receiver adapted to obtain observations of a GNSS signal from        each of a plurality of GNSS satellites;    -   a filter having a state vector comprising a float ambiguity for        each received frequency of the GNSS signals, each float        ambiguity constituting a real number estimate of an integer        number of wavelengths of the GNSS signal between a receiver of        the GNSS signal and the GNSS satellite from which it is        received, the filter estimating a float value for each float        ambiguity of the state vector and covariance values associated        with the state vector;    -   a processing element adapted to        -   assign integer values to at least a subgroup of the            estimated float values to define a plurality of integer            ambiguity candidate sets;        -   obtain a weighted average of the candidate sets;        -   determine a formal precision value based on covariance            values of the filter, the formal precision value being a            measure for an achievable precision;        -   determine an achieved precision value of the weighted            average;        -   compare the achieved precision value with the formal            precisimi value to obtain a convergence value; and        -   indicate a convergence of the determination of the state            vector.-   25. Apparatus of 24, wherein the processing element is adapted to    obtain the convergence value as a ratio of the achieved precision    value to the formal precision value.-   26. Apparatus of at least one of 24 and 25, wherein the processing    element is adapted to    -   determine an instance in time when the convergence value of the        position is better than a convergence threshold; and    -   indicate a convergence of the determination of the state vector        at and after the determined instance in time.-   27. Apparatus of at least one of 24 to 26, wherein the processing    element is adapted to    -   estimate an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicate a convergence of the determination of the state vector,        if the achieved precision of the position is better than an        inclusion threshold-   28. Apparatus of at least one of 24 to 27, wherein the processing    element is adapted to    -   estimate an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicate a non-convergence of the determination of the state        vector, if the achieved precision of the position is worse than        an exclusion threshold.-   29. Apparatus of at least one of 24 to 28, wherein the processing    element is adapted to, in order to obtain the weighted average of    the candidate sets;    -   select a first number of candidate sets having a quality measure        better than a first threshold, wherein the first threshold is        determined based on a reference quality measure of a reference        candidate set; and    -   form the weighted average of the selected candidate sets by        weighing each candidate set in the weighted average based on its        quality measure.-   30. Apparatus of 29, wherein the processing element is adapted to    use the weighted average to estimate a position of the receiver of    the GNSS signals.-   31. Apparatus of at least one of 29 and 30, wherein the reference    candidate set is the candidate set having the best quality measure.-   32. Apparatus of at least one of 29 to 31, wherein the quality    measure of the candidate sets is constituted by a residual error    norm value, the residual error norm value of a candidate set being a    measure for a statistical distance of the candidate set to the state    vector having the float ambiguities.-   33. Apparatus of at least one of 29 to 32, wherein the processing    element is adapted to determine the first threshold as at least one    of a fraction of, a multiple of, and a distance to the reference    quality measure.-   34. Apparatus of at least one of 29 to 33, wherein the processing    element is adapted to, if the first number of selected candidate    sets is smaller than a second threshold, select, for forming the    weighted average, a second number of further candidate sets on the    basis of the quality measures of the candidate sets in decreasing    order starting with the non-selected candidate set having the best    quality measure, the second number being constituted by the    difference between the first number of selected candidate sets and a    second threshold defining a minimum number of candidate sets to be    included in the weighted average.-   35. Apparatus of at least one of 29 to 34, wherein the processing    element is adapted to, if the first number of selected candidate    sets is larger than a third threshold, exclude a third number of    selected candidate sets from forming the weighted average in    decreasing order starting with the selected candidate set having the    worst quality measure, wherein the third number is constituted by    the difference between the first number of selected candidate sets    and a third threshold defining a maximum number of candidate sets to    be included in the weighted average.-   36. Apparatus of at least one of 24 to 35, wherein the processing    element is adapted to, in order to obtain the weighted average:    -   determine the best quality measure of the candidate sets;    -   determine an expectation value of the candidate set having the        best quality measure;    -   determine an error measure as a ratio of the best quality        measure to the expectation value;    -   adapt the quality measures of the candidate sets as a function        of the error measure;    -   perform a selection of the first number of candidate sets based        on the adapted quality measures; and    -   form the weighted average on the basis of the quality measures.-   37. Apparatus of at least one of 24 to 36, wherein the processing    element is adapted to, in order to obtain the weighted average:    -   determine the best quality measure of the candidate sets;    -   determine an expectation value of the candidate set having the        best quality measure;    -   determine an error measure as a ratio of the best quality        measure to the expectation value;    -   adapt the quality measures of the candidate sets as a function        of the error measure; and    -   form the weighted average based on the adapted quality measures.-   38. Apparatus of at least one of 36 and 37, wherein the processing    element is adapted to adapt the quality measures of the candidate    sets as a function of the error measure by adjusting the    variance-covariance matrix of the float solution.-   39. Apparatus of at least one of 36 to 38, wherein the processing    element is adapted to adjust a variance-covariance matrix of the    filter using the error measure, if the error measure is in a    predetermined range.-   40. Apparatus of at least one of 36 to 39, wherein the processing    element is adapted to adjust a variance-covariance matrix of the    filter using the error measure, if the error measure is larger than    one.-   41. Apparatus of at least one of 24 to 40, wherein the processing    element is adapted to, so as to define the integer candidate sets:    -   select a subset of float ambiguities of the state vector to form        the subgroup of the float ambiguities of the state vector used        to define the plurality of integer ambiguity candidate sets; and    -   assign integer values to the estimated float values of the float        ambiguities of the subset to define a plurality of integer        ambiguity candidate sets.-   42. Apparatus of 41, wherein the processing element is adapted to    select, as float ambiguities of the subset, the float ambiguities of    frequencies tracked continuously for the longest period of time.-   43. Apparatus of at least one of 24 to 42, adapted to    -   obtain observations of the at least one frequency of the GNSS        signals from the plurality of GNSS satellites to obtain        observations at a plurality of instances in time;    -   update the float ambiguities of the state vector over time on        the basis of the observations;    -   determine that an interruption in tracking of at least one        signal of a satellite occurred; and    -   maintain the float ambiguity of the state vector for the at        least one signal for which an interruption in tracking occurred        at the value before the interruption in tracking occurred.-   44. Apparatus of 43, adapted to determine that an interruption in    tracking of at least one signal of a satellite occurred if an    observation for the at least one signal is not available for at    least one of the instances in time.-   45. Apparatus of at least one of 43 and 44, adapted to determine    that an interruption in tracking of at least one signal of a    satellite occurred, if a cycle slip occurred.-   46. Apparatus of at least one of 43 to 45 adapted to, if, after an    interruption in tracking of a signal, tracking of the signal    resumes, maintain the float ambiguity of the state vector for the    signal for which an interruption in tracking occurred at the value    before the interruption in tracking occurred as a first float    ambiguity and introducing into the state vector a second float    ambiguity for the signal after resuming tracking.-   47. Rover including an apparatus according to any one of 24 to 46.-   48. Network station including an apparatus according to any one of    24 to 46.-   49. Computer program comprising instructions configured, when    executed on a computer processing unit, to carry out a method    according to any one of 1 to 23.-   50. Computer-readable medium comprising a computer program according    to 49.

(4^(th) Aspect: Keeping Legacy Observations after Interruption ofTracking)

-   1. Method to estimate parameters derived from global navigational    satellite system (GNSS) signals useful to determine a position,    including    -   obtaining observations of each of received frequencies of a GNSS        signal from a plurality of GNSS satellites to obtain        observations at a plurality of instances in time;    -   feeding the time sequence of observations to a filter to        estimate a state vector at least comprising float ambiguities,        wherein each float ambiguity constitutes a real number estimate        of an integer number of wavelengths for a received frequency of        a GNSS signal between a receiver of the GNSS signal and the GNSS        satellite from which it is received and wherein the float        ambiguities of the state vector are updated over time on the        basis of the observations;    -   determining that an interruption in tracking of at least one        signal of a satellite occurred;    -   maintaining the float ambiguity of the state vector for the at        least one signal for which an interruption in tracking occurred        at the value before the interruption in tracking occurred;    -   assigning integer values to at least a subgroup of the estimated        float values to define a plurality of integer ambiguity        candidate sets;    -   determining a quality measure for each of the candidate sets;        and    -   obtaining a weighted average of the candidate sets.-   2. Method of 1, wherein it is determined that an interruption in    tracking of at least one signal of a satellite occurred if an    observation for the at least one signal is not available for at    least one of the instances in time.-   3. Method of at least one of 1 and 2, wherein it is determined that    an interruption in tracking of at least one signal of a satellite    occurred, if a cycle slip occurred.-   4. Method of at least one of 1 to 3 including, if, after an    interruption in tracking of a signal, tracking of the signal    resumes, maintaining the float ambiguity of the state vector for the    signal for which an interruption in tracking occurred at the value    before the interruption in tracking occurred as a first float    ambiguity and introducing into the state vector a second float    ambiguity for the signal after resuming tracking.-   5. Method of at least one of 1 to 5, wherein obtaining the weighted    average includes    -   selecting a first number of candidate sets having a quality        measure better than a first threshold, wherein the first        threshold is determined based on a reference quality measure a        reference candidate set; and    -   obtaining the weighted average of the selected candidate sets by        weighing each candidate set in the weighted average based on its        quality measure.-   6. Method of 5, including using the weighted average to estimate a    position of the receiver of the GNSS signals.-   7. Method of at least one of 5 and 6, wherein the reference    candidate set is the candidate set having the best quality measure.-   8. Method of at least one of 5 to 7, wherein the quality measure of    the candidate sets is constituted by a residual error norm value,    the residual error norm value of a candidate set being a measure for    a statistical distance of the candidate set to the state vector    having the float ambiguities.-   9. Method of at least one of 5 to 8, wherein the first threshold is    determined as at least one of a fraction of, a multiple of, and a    distance to the reference quality measure.-   10. Method of at least one of 5 to 9, including, if the first number    of selected candidate sets is smaller than a second threshold,    selecting, for forming the weighted average, a second number of    further candidate sets on the basis of the quality measures of the    candidate sets in decreasing order starting with the non-selected    candidate set having the best quality measure, the second number    being constituted by the difference between the first number of    selected candidate sets and a second threshold defining a minimum    number of candidate sets to be included in the weighted average.-   11. Method of at least one of 5 to 10, including, if the first    number of selected candidate sets is larger than a third threshold,    excluding a third number of selected candidate sets from forming the    weighted average in decreasing order starting with the selected    candidate set having the worst quality measure, wherein the third    number is constituted by the difference between the first number of    selected candidate sets and a third threshold defining a maximum    number of candidate sets to be included in the weighted average.-   12. Method of at least one of 1 to 11, wherein obtaining the    weighted average includes:    -   determining the best quality measure of the candidate sets;    -   determining an expectation value of the candidate set having the        best quality measure;    -   determining an error measure as a ratio of the best quality        measure to the expectation value;    -   adapting the quality measures of the candidate sets as a        function of the error measure;    -   performing a selection of the first number of candidate sets        based on the adapted quality measures; and    -   forming the weighted average on the basis of the quality        measures.-   13. Method of at least one of 1 to 12, wherein obtaining the    weighted average includes:    -   determining the best quality measure of the candidate sets;    -   determining an expectation value of the candidate set having the        best quality measure;    -   determining an error measure as a ratio of the best quality        measure to the expectation value;    -   adapting the quality measures of the candidate sets as a        function of the error measure; and    -   forming the weighted average based on the adapted quality        measures.-   14. Method of at least one of 1 to 13, including adapting the    quality measures of the candidate sets as a function of the error    measure by adjusting a variance-covariance matrix of the float    solution so as to change the distribution of the quality measures of    the candidate sets relative to the float solution.-   15. Method of at least one of 12 to 14, including adjusting a    variance-covariance matrix of the filter using the error measure, if    the error measure is in a predetermined range.-   16. Method of at least one of 12 to 15, including adjusting a    variance-covariance matrix of the filter using the error measure, if    the error measure is larger than one.-   17. Method of at least one of 1 to 16, wherein defining the integer    candidate sets includes:    -   selecting a subset of float ambiguities of the state vector to        form the subgroup of the float ambiguities of the state vector        used to define the plurality of integer ambiguity candidate        sets; and    -   assigning integer values to the estimated float values of the        float ambiguities of the subset to define the plurality of        integer ambiguity candidate sets.-   18. Method of 17, wherein the selecting includes selecting, as float    ambiguities of the subset, the float ambiguities of frequencies    tracked continuously for the longest period of time.-   19. Method of at least one 1 to 18, including    -   estimating by the filter a float value for each float ambiguity        of the state vector and covariance values associated with the        state vector;    -   determining a formal precision value based on covariance values        of the filter, the formal precision value being a measure for an        achievable precision;    -   determining an achieved precision value of the weighted average;    -   comparing the achieved precision value with the formal precision        value to obtain a convergence value; and    -   indicating a convergence of the determination of the state        vector based on the convergence value.-   20. Method of 19, wherein the convergence value is obtained as a    ratio of the achieved precision value to the formal precision value.-   21. Method of at least one of 19 and 20, including    -   determining an instance in time when the convergence value of        the position is better than a convergence threshold; and    -   indicating a convergence of the determination of the state        vector at and after the determined instance in time.-   22. Method of at least one of 19 to 21, including    -   estimating an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicating a convergence of the determination of the state        vector, if the achieved precision of the position is better than        an inclusion threshold-   23. Method of at least one of 19 to 22, including    -   estimating an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicating a non-convergence of the determination of the state        vector, if the achieved precision of the position is worse than        an exclusion threshold.-   24. Apparatus to estimate parameters derived from global    navigational satellite system (GNSS) signals useful to determine a    position, including    -   a receiver adapted to obtain observations of each of received        frequencies of a GNSS signal from a plurality of GNSS satellites        to obtain observations at a plurality of instances in time;    -   a filter to estimate a state vector at least comprising float        ambiguities based on the time sequence of observations, wherein        each float ambiguity constitutes a real number estimate of an        integer number of wavelengths for a received frequency of a GNSS        signal between a receiver of the GNSS signal and the GNSS        satellite from which it is received and wherein the float        ambiguities of the state vector are updated over time on the        basis of the observations;    -   a processing element adapted to        -   determine that an interruption in tracking of at least one            signal of a satellite occurred;        -   maintain the float ambiguity of the state vector for the at            least one signal for which an interruption in tracking            occurred at the value before the interruption in tracking            occurred;        -   assign integer values to at least a subgroup of the            estimated float values to define a plurality of integer            ambiguity candidate sets;        -   determine a quality measure for each of the candidate sets;            and        -   obtain a weighted average of the candidate sets.-   25. Apparatus of 24, wherein the processing element is adapted to    determine that an interruption in tracking of at least one signal of    a satellite occurred if an observation for the at least one signal    is not available for at least one of the instances in time.-   26. Apparatus of at least one of 24 and 25, wherein the processing    element is adapted to determine that an interruption in tracking of    at least one signal of a satellite occurred, if a cycle slip    occurred.-   27. Apparatus of at least one of 24 to 26 wherein the processing    element is adapted to, if, after an interruption in tracking of a    signal, tracking of the signal resumes, maintain the float ambiguity    of the state vector for the signal for which an interruption in    tracking occurred at the value before the interruption in tracking    occurred as a first float ambiguity and introducing into the state    vector a second float ambiguity for the signal after resuming    tracking.-   28. Apparatus of at least one of 24 to 27, wherein the processing    element is adapted to, in order to obtain the weighted average    -   select a first number of candidate sets having a quality measure        better than a first threshold, wherein the first threshold is        determined based on a reference quality measure a reference        candidate set; and    -   obtain the weighted average of the selected candidate sets by        weighing each candidate set in the weighted average based on its        quality measure.-   29. Apparatus of 28, wherein the processing element is adapted to    use the weighted average to estimate a position of the receiver of    the GNSS signals.-   30. Apparatus of at least one of 28 and 29, wherein the reference    candidate set is the candidate set having the best quality measure.-   31. Apparatus of at least one of 28 to 30, wherein the quality    measure of the candidate sets is constituted by a residual error    norm value, the residual error norm value of a candidate set being a    measure for a statistical distance of the candidate set to the state    vector having the float ambiguities.-   32. Apparatus of at least one of 28 to 31, wherein the processing    element is adapted to determine the first threshold as at least one    of a fraction of, a multiple of, and a distance to the reference    quality measure.-   33. Apparatus of at least one of 28 to 32, wherein the processing    element is adapted to, if the first number of selected candidate    sets is smaller than a second threshold, select, for forming the    weighted average, a second number of further candidate sets on the    basis of the quality measures of the candidate sets in decreasing    order starting with the non-selected candidate set having the best    quality measure, the second number being constituted by the    difference between the first number of selected candidate sets and a    second threshold defining a minimum number of candidate sets to be    included in the weighted average.-   34. Apparatus of at least one of 28 to 33, wherein the processing    element is adapted to, if the first number of selected candidate    sets is larger than a third threshold, exclude a third number of    selected candidate sets from forming the weighted average in    decreasing order starting with the selected candidate set having the    worst quality measure, wherein the third number is constituted by    the difference between the first number of selected candidate sets    and a third threshold defining a maximum number of candidate sets to    be included in the weighted average.-   35. Apparatus of at least one of 24 to 34, wherein the processing    element is adapted to, in order to obtain the weighted average:    -   determine the best quality measure of the candidate sets;    -   determine an expectation value of the candidate set having the        best quality measure;    -   determine an error measure as a ratio of the best quality        measure to the expectation value;    -   adapt the quality measures of the candidate sets as a function        of the error measure;    -   perform a selection of the first number of candidate sets based        on the adapted quality measures; and    -   form the weighted average on the basis of the quality measures.-   36. Apparatus of at least one of 24 to 35, wherein the processing    element is adapted to, in order to obtain the weighted average:    -   determine the best quality measure of the candidate sets;    -   determine an expectation value of the candidate set having the        best quality measure;    -   determine an error measure as a ratio of the best quality        measure to the expectation value;    -   adapt the quality measures of the candidate sets as a function        of the error measure; and    -   form the weighted average based on the adapted quality measures.-   37. Apparatus of at least one of 24 to 36, wherein the processing    element is adapted to adapt the quality measures of the candidate    sets as a function of the error measure by adjusting a    variance-covariance matrix of the float solution so as to change the    distribution of the quality measures of the candidate sets relative    to the float solution.-   38. Apparatus of at least one of 35 to 37, wherein the processing    element is adapted to adjust a variance-covariance matrix of the    filter using the error measure, if the error measure is in a    predetermined range.-   39. Apparatus of at least one of 35 to 38, wherein the processing    element is adapted to adjust a variance-covariance matrix of the    filter using the error measure, if the error measure is larger than    one.-   40. Apparatus of at least one of 24 to 39, wherein the processing    element is adapted to, in order to define the integer candidate    sets:    -   select a subset of float ambiguities of the state vector to form        the subgroup of the float ambiguities of the state vector used        to define the plurality of integer ambiguity candidate sets; and    -   assign integer values to the estimated float values of the float        ambiguities of the subset to define the plurality of integer        ambiguity candidate sets.-   41. Apparatus of 40, wherein the processing element is adapted to    select, as float ambiguities of the subset, the float ambiguities of    frequencies tracked continuously for the longest period of time.-   42. Apparatus of at least one 24 to 41, wherein the filter is    adapted to    -   estimate by the filter a float value for each float ambiguity of        the state vector and covariance values associated with the state        vector; and    -   wherein the processing element is adapted to        -   determine a formal precision value based on covariance            values of the filter, the formal precision value being a            measure for an achievable precision;        -   determine an achieved precision value of the weighted            average;        -   compare the achieved precision value with the formal            precision value to obtain a convergence value; and        -   indicate a convergence of the determination of the state            vector based on the convergence value.-   43. Apparatus of 42, wherein the processing element is adapted to    obtain the convergence value as a ratio of the achieved precision    value to the formal precision value.-   44. Apparatus of at least one of 42 and 43, wherein the processing    element is adapted to    -   determine an instance in time when the convergence value of the        position is better than a convergence threshold; and    -   indicate a convergence of the determination of the state vector        at and after the determined instance in time.-   45. Apparatus of at least one of 42 to 44, wherein the processing    element is adapted to    -   estimate an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicate a convergence of the determination of the state vector,        if the achieved precision of the position is better than an        inclusion threshold-   46. Apparatus of at least one of 42 to 45, wherein the processing    element is adapted to    -   estimate an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicate a non-convergence of the determination of the state        vector, if the achieved precision of the position is worse than        an exclusion threshold.-   47. Rover including an apparatus according to any one of 24 to 46.-   48. Network station including an apparatus according to any one of    24 to 46.-   49. Computer program comprising instructions configured, when    executed on a computer processing unit, to carry out a method    according to any one of 1 to 23.-   50. Computer-readable medium comprising a computer program according    to 49.

(5^(th) Aspect: Ambiguity Selection)

-   1. Method to estimate parameters derived from global navigational    satellite system (GNSS) signals useful to determine a position,    comprising    -   obtaining observations of a GNSS signal from each of a plurality        of GNSS satellites;    -   feeding the observations to a filter having a state vector at        least comprising a float ambiguity for each received frequency        of the GNSS signals, each float ambiguity constituting a real        number estimate associated with an integer number of wavelengths        of the GNSS signal between a receiver of the GNSS signal and the        GNSS satellite from which it is received, and the filter being        for estimating a float value for each float ambiguity of the        state vector;    -   selecting a subset of the ambiguities of the state vector;    -   assigning integer values to the estimated float values of the        float ambiguities of the subset to define a plurality of        candidate sets;    -   determining a quality measure for each of the candidate sets;        and    -   forming a weighted average of the candidate sets.-   2. The method of 1, wherein selecting a subset of float ambiguities    of the state vector comprises one of: (a) operating the filter to    estimate from the observations a complete set of float ambiguity    values and subsequently selecting the subset of float ambiguities    from among the complete set of float ambiguity values, and (b)    operating the filter to estimate from the observations a selected    partial set of float ambiguity values comprising fewer than the    complete set of float ambiguity values, wherein the partial set of    float ambiguity values comprises the subset of float ambiguities of    the state vector.-   3. Method of 1 or 2, wherein the selecting comprises selecting, as    float ambiguities of the subset, the float ambiguities corresponding    to frequencies of satellites tracked continuously for the longest    period of time.-   4. Method of at least one of 1 to 3, wherein forming the weighted    average comprises    -   selecting a first number of candidate sets having a quality        measure better than a first threshold, wherein the first        threshold is determined based on a reference quality measure of        a reference candidate set; and    -   forming a weighted average of the selected candidate sets, each        candidate set weighted in the weighted average based on its        quality measure.-   5. Method of at least one of 1 to 4, further comprising:    -   determining a problem in the floating solution by at least one        of (1) comparing a quality measure of a reference candidate set        with a problem threshold and (2) determining an improvement        factor by comparing the quality measure of the reference        candidate set with a quality measure of a superset of the float        ambiguities of the reference candidate set, and    -   resetting the estimated float value of at least one of the float        ambiguities which is not included in the candidate set.-   6. Method of one of 1 to 5, comprising using the weighted average to    estimate a position of the receiver of the GNSS signals.-   7. Method of at least one of 4 and 5, wherein the reference    candidate set is the candidate set having the best quality measure.-   8. Method of at least one of 4 to 7, wherein the quality measure of    the candidate sets is constituted by a residual error norm value,    the residual error norm value of a candidate set being a measure for    a statistical distance of the candidate set to the state vector    having the float ambiguities.-   9. Method of at least one of 4 to 8, wherein the first threshold is    determined as at least one of a fraction of, a multiple of, and a    distance to the reference quality measure.-   10. Method of at least one of 4 to 9, comprising, if the first    number of selected candidate sets is smaller than a second    threshold, selecting, for forming the weighted average, a second    number of further candidate sets on the basis of the quality    measures of the candidate sets in decreasing order starting with the    non-selected candidate set having the best quality measure, the    second number being constituted by the difference between the first    number of selected candidate sets and a second threshold defining a    minimum number of candidate sets to be included in the weighted    average.-   11. Method of at least one of 4 to 10 comprising, if the first    number of selected candidate sets is larger than a third threshold,    excluding a third number of selected candidate sets from forming the    weighted average in decreasing order starting with the selected    candidate set having the worst quality measure, wherein the third    number is constituted by the difference between the first number of    selected candidate sets and a third threshold defining a maximum    number of candidate sets to be included in the weighted average.-   12. Method of at least one of 1 to 11, wherein forming the weighted    average comprises:    -   determining the best quality measure of the candidate sets;    -   determining an expectation value of the candidate set having the        best quality measure;    -   determining an error measure as a ratio of the best quality        measure to the expectation value;    -   adapting the quality measures of the candidate sets as a        function of the error measure; and    -   performing a selection of a first number of candidate sets based        on the adapted quality measures.-   13. Method of at least one of 1 to 12, wherein forming the weighted    average comprises:    -   determining the best quality measure of the candidate sets;    -   determining an expectation value of the candidate set having the        best quality measure;    -   determining an error measure as a ratio of the best quality        measure to the expectation value;    -   adapting the quality measures of the candidate sets as a        function of the error measure; and    -   forming the weighted average based on the adapted quality        measure.-   14. Method of at least one of 12 and 13, comprising adapting the    quality measures of the candidate sets as a function of the error    measure by adjusting the variance-covariance matrix of the float    solution.-   15. Method of at least one of 12 to 14, comprising adjusting a    variance-covariance matrix of the filter using the error measure, if    the error measure is in a predetermined range.-   16. Method of at least one of 12 to 14, comprising adjusting a    variance-covariance matrix of the filter using the error measure, if    the error measure is larger than one.-   17. Method of at least one of 1 to 16, comprising    -   estimating by the filter a float value for each float ambiguity        of the state vector and covariance values associated with the        state vector;    -   determining a formal precision value based on covariance values        derived of the filter, the formal precision value being a        measure for an achievable precision;    -   determining an achieved precision value of the weighted average;        comparing the achieved precision value with the formal precision        value to obtain a convergence value; and    -   indicating a convergence of the determination of the state        vector based on the convergence value.-   18. Method of 17, wherein the convergence value is obtained as a    ratio of the achieved precision value to the formal precision value.-   19. Method of at least one of 17 and 18, comprising    -   determining an instance in time when the convergence value of        the position is better than a convergence threshold; and    -   indicating a convergence of the determination of the state        vector at and after the determined instance in time.-   20. Method of at least one of 17 to 19, comprising    -   estimating an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicating a convergence of the determination of the state        vector, if the achieved precision of the position is better than        an inclusion threshold.-   21. Method of at least one of 17 to 20, comprising    -   estimating an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicating a non-convergence of the determination of the state        vector, if the achieved precision of the position is worse than        an exclusion threshold.-   22. Method of at least one of 1 to 21, comprising:    -   obtaining observations of the at least one frequency of the GNSS        signals from the plurality of GNSS satellites to obtain        observations at a plurality of instances in time;    -   updating the float ambiguities of the state vector over time on        the basis of the observations;    -   determining that an interruption in tracking of at least one        signal of a satellite occurred; and    -   maintaining the float ambiguity of the state vector for the at        least one signal for which an interruption in tracking occurred        at the value before the interruption in tracking occurred.-   23. Method of 22, wherein it is determined that an interruption in    tracking of at least one signal of a satellite occurred if an    observation for the at least one signal is not available for at    least one of the instances in time.-   24. Method of at least one of 22 and 23, wherein it is determined    that an interruption in tracking of at least one signal of a    satellite occurred, if a cycle slip occurred.-   25. Method of at least one of 22 to 24 comprising, if, after an    interruption in tracking of a signal, tracking of the signal    resumes, maintaining the float ambiguity of the state vector for the    signal for which an interruption in tracking occurred at the value    before the interruption in tracking occurred as a first float    ambiguity and introducing into the state vector a second float    ambiguity for the signal after resuming tracking.-   26. Apparatus to estimate parameters derived from global    navigational satellite system (GNSS) signals useful to determine a    position, comprising    -   a receiver adapted to obtain observations of a GNSS signal from        each of a plurality of GNSS satellites;    -   a filter having a state vector at least comprising a float        ambiguity for each received frequency of the GNSS signals, each        float ambiguity constituting a real number estimate associated        with an integer number of wavelengths of the GNSS signal between        a receiver of the GNSS signal and the GNSS satellite from which        it is received, and the filter being for estimating a float        value for each float ambiguity of the state vector;    -   a processing element adapted to        -   select a subset of float ambiguities of the state vector;        -   assign integer values to the estimated float values of the            float ambiguities of the subset to    -   define a plurality of integer ambiguity candidate sets;        -   determine a quality measure for each of the candidate sets;            and        -   form a weighted average of the candidate sets.-   27. Apparatus of 26, wherein the processing is adapted to select a    subset of float ambiguities of the state vector by one of: (a)    operating the filter to estimate from the observations a complete    set of float ambiguity values and subsequently selecting the subset    of float ambiguities from among the complete set of float ambiguity    values, and (b) operating the filter to estimate from the    observations a selected partial set of float ambiguity values    comprising fewer than the full set of float ambiguity values,    wherein the partial set of float ambiguity values comprises the    subset of float ambiguities of the state vector.-   28. Apparatus of at least one of 26 and 27, wherein the processing    element is adapted to select, as float ambiguities of the subset,    the float ambiguities corresponding to frequencies of satellites    tracked continuously for the longest period of time.-   29. Apparatus of at least one of 26 to 28, wherein the processing    element is adapted to, in order to form the weighted average,    -   select a first number of candidate sets having a quality measure        better than a first threshold, wherein the first threshold is        determined based on a reference quality measure a reference        candidate set; and    -   form a weighted average of the selected candidate sets, each        candidate set weighted in the weighted average based on its        quality measure.-   30. Apparatus of at least one of 26 to 29, wherein the processing    element is further adapted to:    -   determine a problem in the floating solution by at least one        of (1) comparing a quality measure of a reference candidate set        with a problem threshold and (2) determining an improvement        factor by comparing the quality measure of the reference        candidate set with a quality measure of a superset of the float        ambiguities of the reference candidate set, and    -   reset the estimated float value of at least one of the float        ambiguities which is not included in the candidate set.-   31. Apparatus of at least one of 26 to 29, wherein the processing    element is adapted to use the weighted average to estimate a    position of the receiver of the GNSS signals.-   32. Apparatus of at least one of 29 and 30, wherein the reference    candidate set is the candidate set having the best quality measure.-   33. Apparatus of at least one of 29 to 32, wherein the quality    measure of the candidate sets is constituted by a residual error    norm value, the residual error norm value of a candidate set being a    measure for a statistical distance of the candidate set to the state    vector having the float ambiguities.-   34. Apparatus of at least one of 29 to 32, wherein the processing    element is adapted to determine the first threshold as at least one    of a fraction of, a multiple of, and a distance to the reference    quality measure.-   35. Apparatus of at least one of 29 to 34, wherein the processing    element is adapted to, if the first number of selected candidate    sets is smaller than a second threshold, select, for forming the    weighted average, a second number of further candidate sets on the    basis of the quality measures of the candidate sets in decreasing    order starting with the non-selected candidate set having the best    quality measure, the second number being constituted by the    difference between the first number of selected candidate sets and a    second threshold defining a minimum number of candidate sets to be    included in the weighted average.-   36. Apparatus of at least one of 29 to 35, wherein the processing    element is adapted to, if the first number of selected candidate    sets is larger than a third threshold, exclude a third number of    selected candidate sets from forming the weighted average in    decreasing order starting with the selected candidate set having the    worst quality measure, wherein the third number is constituted by    the difference between the first number of selected candidate sets    and a third threshold defining a maximum number of candidate sets to    be included in the weighted average.-   37. Apparatus of at least one of 26 to 36, wherein the processing    element is adapted to, in order to form the weighted average,    -   determine the best quality measure of the candidate sets;    -   determine an expectation value of the candidate set having the        best quality measure;    -   determine an error measure as a ratio of the best quality        measure to the expectation value;    -   adapt the quality measures of the candidate sets as a function        of the error measure; and    -   perform a selection of a first number of candidate sets based on        the adapted quality measures.-   38. Apparatus of at least one of 26 to 37, wherein the processing    element is adapted to, in order to form the weighted average,    -   determine the best quality measure of the candidate sets;    -   determine an expectation value of the candidate set having the        best quality measure;    -   determine an error measure as a ratio of the best quality        measure to the expectation value;    -   adapt the quality measures of the candidate sets as a function        of the error measure; and    -   form the weighted average based on the adapted quality measure.-   39. Apparatus of at least one of 37 and 38, wherein the processing    element is adapted to adapt the quality measures of the candidate    sets as a function of the error measure by adjusting the    variance-covariance matrix of the float solution.-   40. Apparatus of at least one of 37 to 39, wherein the processing    element is adapted to adjust a variance-covariance matrix of the    filter using the error measure, if the error measure is in a    predetermined range.-   41. Apparatus of at least one of 37 to 40, wherein the processing    element is adapted to adjust a variance-covariance matrix of the    filter using the error measure, if the error measure is larger than    one.-   42. Apparatus of at least one of 26 to 41, wherein the filter is    adapted to estimate a float value for each float ambiguity of the    state vector and covariance values associated with the state vector;    and the processing element is adapted to    -   determine a formal precision value based on covariance values        derived of the filter, the formal precision value being a        measure for an achievable precision;    -   determine an achieved precision value of the weighted average;    -   compare the achieved precision value with the formal precision        value to obtain a convergence value; and    -   indicate a convergence of the determination of the state vector        based on the convergence value.-   43. Apparatus of 42, wherein the processing element is adapted to    obtain the convergence value as a ratio of the achieved precision    value to the formal precision value.-   44. Apparatus of at least one of 42 and 43, wherein the processing    element is adapted to    -   determine an instance in time when the convergence value of the        position is better than a convergence threshold; and    -   indicate a convergence of the determination of the state vector        at and after the determined instance in time.-   45. Apparatus of at least one of 42 to 44, wherein the processing    element is adapted to    -   estimate an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicate a convergence of the determination of the state vector,        if the achieved precision of the position is better than an        inclusion threshold.-   46. Apparatus of at least one of 42 to 45, wherein the processing    element is adapted to    -   estimate an achieved precision of a position of the receiver        determined based on the weighted ambiguities; and    -   indicate a non-convergence of the determination of the state        vector, if the achieved precision of the position is worse than        an exclusion threshold.-   47. Apparatus of at least one of 26 to 46, wherein the filter is    adapted to obtain observations of the at least one frequency of the    GNSS signals from the plurality of GNSS satellites to obtain    observations at a plurality of instances in time; and the processing    element is adapted to    -   update the float ambiguities of the state vector over time on        the basis of the observations;    -   determine that an interruption in tracking of at least one        signal of a satellite occurred; and    -   maintain the float ambiguity of the state vector for the at        least one signal for which an interruption in tracking occurred        at the value before the interruption in tracking occurred.-   48. Apparatus of 47, wherein the processing element is adapted to    determine that an interruption in tracking of at least one signal of    a satellite occurred if an observation for the at least one signal    is not available for at least one of the instances in time.-   49. Apparatus of at least one of 47 and 48, wherein the processing    element is adapted to determine that an interruption in tracking of    at least one signal of a satellite occurred, if a cycle slip    occurred.-   50. Apparatus of at least one of 47 to 49 wherein the processing    element is adapted to, if, after an interruption in tracking of a    signal, tracking of the signal resumes, maintain the float ambiguity    of the state vector for the signal for which an interruption in    tracking occurred at the value before the interruption in tracking    occurred as a first float ambiguity and introducing into the state    vector a second float ambiguity for the signal after resuming    tracking.-   51. Rover comprising an apparatus according to any one of 26 to 50.-   52. Network station comprising an apparatus according to any one of    26 to 51.-   53. Computer program comprising instructions configured, when    executed on a computer processing unit, to carry out a method    according to any one of 1 to 25.-   54. Computer-readable medium comprising one of: a computer-readable    physical storage medium embodying a computer program according to 53    and a computer-readable transmission medium embodying a computer    program according to 53.

1. Method to estimate parameters derived from global navigationalsatellite system (GNSS) signals useful to determine a position,comprising obtaining observations of a GNSS signal from each of aplurality of GNSS satellites; feeding the observations to a filterhaving a state vector comprising a float ambiguity for each receivedfrequency of the GNSS signals, each float ambiguity constituting a realnumber estimate of an integer number of wavelengths of the GNSS signalbetween a receiver of the GNSS signal and the GNSS satellite from whichit is received, the filter estimating a float value for each floatambiguity of the state vector and covariance values associated with thestate vector; assigning integer values to at least a subgroup of theestimated float values to define a plurality of integer ambiguitycandidate sets; obtaining a weighted average of the candidate sets;determining a formal precision value based on covariance values of thefilter, the formal precision value being a measure for an achievableprecision; determining an achieved precision value of the weightedaverage; comparing the achieved precision value with the formalprecision value to obtain a convergence value; and indicating aconvergence of the determination of the state vector.
 2. Method of claim1, wherein the convergence value is obtained as a ratio of the achievedprecision value to the formal precision value.
 3. Method of claim 1,comprising determining an instance in time when the convergence value ofthe position is better than a convergence threshold; and indicating aconvergence of the determination of the state vector at and after thedetermined instance in time.
 4. Method of claim 1, comprising estimatingan achieved precision of a position of the receiver determined based onthe weighted ambiguities; and indicating a convergence of thedetermination of the state vector, if the achieved precision of theposition is better than an inclusion threshold.
 5. Method of at claim 1,comprising estimating an achieved precision of a position of thereceiver determined based on the weighted ambiguities; and indicating anon-convergence of the determination of the state vector, if theachieved precision of the position is worse than an exclusion threshold.6. Method of claim 1, wherein obtaining a weighted average of thecandidate sets comprises; selecting a first number of candidate setshaving a quality measure better than a first threshold, wherein thefirst threshold is determined based on a reference quality measure of areference candidate set; and forming the weighted average of theselected candidate sets by weighing each candidate set in the weightedaverage based on its quality measure.
 7. Method of claim 6, comprisingusing the weighted average to estimate a position of the receiver of theGNSS signals.
 8. Method of claim 6, wherein the reference candidate setis the candidate set having the best quality measure.
 9. Method of atclaim 6, wherein the quality measure of the candidate sets isconstituted by a residual error norm value, the residual error normvalue of a candidate set being a measure for a statistical distance ofthe candidate set to the state vector having the float ambiguities. 10.Method of claim 6, wherein the first threshold is determined as at leastone of a fraction of, a multiple of, and a distance to the referencequality measure.
 11. Method of claim 6, comprising, if the first numberof selected candidate sets is smaller than a second threshold,selecting, for forming the weighted average, a second number of furthercandidate sets on the basis of the quality measures of the candidatesets in decreasing order starting with the non-selected candidate sethaving the best quality measure, the second number being constituted bythe difference between the first number of selected candidate sets and asecond threshold defining a minimum number of candidate sets to beincluded in the weighted average.
 12. Method of at claim 6, comprising,if the first number of selected candidate sets is larger than a thirdthreshold, excluding a third number of selected candidate sets fromforming the weighted average in decreasing order starting with theselected candidate set having the worst quality measure, wherein thethird number is constituted by the difference between the first numberof selected candidate sets and a third threshold defining a maximumnumber of candidate sets to be included in the weighted average. 13.Method of claim 1, wherein obtaining the weighted average comprises:determining the best quality measure of the candidate sets; determiningan expectation value of the candidate set having the best qualitymeasure; determining an error measure as a ratio of the best qualitymeasure to the expectation value; adapting the quality measures of thecandidate sets as a function of the error measure; performing aselection of the first number of candidate sets based on the adaptedquality measures; and forming the weighted average on the basis of thequality measures.
 14. Method of claim 1, wherein obtaining the weightedaverage comprises: determining the best quality measure of the candidatesets; determining an expectation value of the candidate set having thebest quality measure; determining an error measure as a ratio of thebest quality measure to the expectation value; adapting the qualitymeasures of the candidate sets as a function of the error measure; andforming the weighted average based on the adapted quality measures. 15.Method of claim 13, comprising adapting the quality measures of thecandidate sets as a function of the error measure by adjusting thevariance-covariance matrix of the float solution.
 16. Method of claim13, comprising adjusting a variance-covariance matrix of the filterusing the error measure, if the error measure is in a predeterminedrange.
 17. Method of claim 13, comprising adjusting avariance-covariance matrix of the filter using the error measure, if theerror measure is larger than one.
 18. Method of claim 1, whereindefining the integer candidate sets comprises: selecting a subset offloat ambiguities of the state vector to form the subgroup of the floatambiguities of the state vector used to define the plurality of integerambiguity candidate sets; and assigning integer values to the estimatedfloat values of the float ambiguities of the subset to define aplurality of integer ambiguity candidate sets.
 19. Method of claim 18,wherein the selecting comprises selecting, as float ambiguities of thesubset, the float ambiguities of frequencies tracked continuously forthe longest period of time.
 20. Method of at at claim 1, comprisingobtaining observations of the at least one frequency of the GNSS signalsfrom the plurality of GNSS satellites to obtain observations at aplurality of instances in time; updating the float ambiguities of thestate vector over time on the basis of the observations; determiningthat an interruption in tracking of at least one signal of a satelliteoccurred; and maintaining the float ambiguity of the state vector forthe at least one signal for which an interruption in tracking occurredat the value before the interruption in tracking occurred.
 21. Method ofclaim 20, wherein it is determined that an interruption in tracking ofat least one signal of a satellite occurred if an observation for the atleast one signal is not available for at least one of the instances intime.
 22. Method of claim 20, wherein it is determined that aninterruption in tracking of at least one signal of a satellite occurred,if a cycle slip occurred.
 23. Method of claim 20 comprising, if, afteran interruption in tracking of a signal, tracking of the signal resumes,maintaining the float ambiguity of the state vector for the signal forwhich an interruption in tracking occurred at the value before theinterruption in tracking occurred as a first float ambiguity andintroducing into the state vector a second float ambiguity for thesignal after resuming tracking.
 24. Apparatus to estimate parametersderived from global navigational satellite system (GNSS) signals usefulto determine a position, comprising a receiver adapted to obtainobservations of a GNSS signal from each of a plurality of GNSSsatellites; a filter having a state vector comprising a float ambiguityfor each received frequency of the GNSS signals, each float ambiguityconstituting a real number estimate of an integer number of wavelengthsof the GNSS signal between a receiver of the GNSS signal and the GNSSsatellite from which it is received, the filter estimating a float valuefor each float ambiguity of the state vector and covariance valuesassociated with the state vector; a processing element adapted to assigninteger values to at least a subgroup of the estimated float values todefine a plurality of integer ambiguity candidate sets; obtain aweighted average of the candidate sets; determine a formal precisionvalue based on covariance values of the filter, the formal precisionvalue being a measure for an achievable precision; determine an achievedprecision value of the weighted average; compare the achieved precisionvalue with the formal precision value to obtain a convergence value; andindicate a convergence of the determination of the state vector. 25.Apparatus of claim 24, wherein the processing element is adapted toobtain the convergence value as a ratio of the achieved precision valueto the formal precision value.
 26. Apparatus of claim 24, wherein theprocessing element is adapted to determine an instance in time when theconvergence value of the position is better than a convergencethreshold; and indicate a convergence of the determination of the statevector at and after the determined instance in time.
 27. Apparatus ofclaim 24, wherein the processing element is adapted to estimate anachieved precision of a position of the receiver determined based on theweighted ambiguities; and indicate a convergence of the determination ofthe state vector, if the achieved precision of the position is betterthan an inclusion threshold.
 28. Apparatus of claim 24, wherein theprocessing element is adapted to estimate an achieved precision of aposition of the receiver determined based on the weighted ambiguities;and indicate a non-convergence of the determination of the state vector,if the achieved precision of the position is worse than an exclusionthreshold.
 29. Apparatus of claim 24, wherein the processing element isadapted to, in order to obtain the weighted average of the candidatesets; select a first number of candidate sets having a quality measurebetter than a first threshold, wherein the first threshold is determinedbased on a reference quality measure of a reference candidate set; andform the weighted average of the selected candidate sets by weighingeach candidate set in the weighted average based on its quality measure.30. Apparatus of claim 29, wherein the processing element is adapted touse the weighted average to estimate a position of the receiver of theGNSS signals.
 31. Apparatus of claims 29, wherein the referencecandidate set is the candidate set having the best quality measure. 32.Apparatus of claim 29, wherein the quality measure of the candidate setsis constituted by a residual error norm value, the residual error normvalue of a candidate set being a measure for a statistical distance ofthe candidate set to the state vector having the float ambiguities. 33.Apparatus of claim 29, wherein the processing element is adapted todetermine the first threshold as at least one of a fraction of, amultiple of, and a distance to the reference quality measure. 34.Apparatus of claim 29, wherein the processing element is adapted to, ifthe first number of selected candidate sets is smaller than a secondthreshold, select, for forming the weighted average, a second number offurther candidate sets on the basis of the quality measures of thecandidate sets in decreasing order starting with the non-selectedcandidate set having the best quality measure, the second number beingconstituted by the difference between the first number of selectedcandidate sets and a second threshold defining a minimum number ofcandidate sets to be included in the weighted average.
 35. Apparatus ofclaim 29, wherein the processing element is adapted to, if the firstnumber of selected candidate sets is larger than a third threshold,exclude a third number of selected candidate sets from forming theweighted average in decreasing order starting with the selectedcandidate set having the worst quality measure, wherein the third numberis constituted by the difference between the first number of selectedcandidate sets and a third threshold defining a maximum number ofcandidate sets to be included in the weighted average.
 36. Apparatus ofclaim 24, wherein the processing element is adapted to, in order toobtain the weighted average: determine the best quality measure of thecandidate sets; determine an expectation value of the candidate sethaving the best quality measure; determine an error measure as a ratioof the best quality measure to the expectation value; adapt the qualitymeasures of the candidate sets as a function of the error measure;perform a selection of the first number of candidate sets based on theadapted quality measures; and form the weighted average on the basis ofthe quality measures.
 37. Apparatus of claim 24, wherein the processingelement is adapted to, in order to obtain the weighted average:determine the best quality measure of the candidate sets; determine anexpectation value of the candidate set having the best quality measure;determine an error measure as a ratio of the best quality measure to theexpectation value; adapt the quality measures of the candidate sets as afunction of the error measure; and form the weighted average based onthe adapted quality measures.
 38. Apparatus of claim 36, wherein theprocessing element is adapted to adapt the quality measures of thecandidate sets as a function of the error measure by adjusting thevariance-covariance matrix of the float solution.
 39. Apparatus of claim36, wherein the processing element is adapted to adjust avariance-covariance matrix of the filter using the error measure, if theerror measure is in a predetermined range.
 40. Apparatus of claim 36,wherein the processing element is adapted to adjust avariance-covariance matrix of the filter using the error measure, if theerror measure is larger than one.
 41. Apparatus of claim 24, wherein theprocessing element is adapted to, so as to define the integer candidatesets: select a subset of float ambiguities of the state vector to formthe subgroup of the float ambiguities of the state vector used to definethe plurality of integer ambiguity candidate sets; and assign integervalues to the estimated float values of the float ambiguities of thesubset to define a plurality of integer ambiguity candidate sets. 42.Apparatus of claim 41, wherein the processing element is adapted toselect, as float ambiguities of the subset, the float ambiguities offrequencies tracked continuously for the longest period of time. 43.Apparatus of claim 24, adapted to obtain observations of the at leastone frequency of the GNSS signals from the plurality of GNSS satellitesto obtain observations at a plurality of instances in time; update thefloat ambiguities of the state vector over time on the basis of theobservations; determine that an interruption in tracking of at least onesignal of a satellite occurred; and maintain the float ambiguity of thestate vector for the at least one signal for which an interruption intracking occurred at the value before the interruption in trackingoccurred.
 44. Apparatus of claim 43, adapted to determine that aninterruption in tracking of at least one signal of a satellite occurredif an observation for the at least one signal is not available for atleast one of the instances in time.
 45. Apparatus of claim 43, adaptedto determine that an interruption in tracking of at least one signal ofa satellite occurred, if a cycle slip occurred.
 46. Apparatus of claim43 adapted to, if, after an interruption in tracking of a signal,tracking of the signal resumes, maintain the float ambiguity of thestate vector for the signal for which an interruption in trackingoccurred at the value before the interruption in tracking occurred as afirst float ambiguity and introducing into the state vector a secondfloat ambiguity for the signal after resuming tracking.
 47. Rovercomprising an apparatus according to claim
 24. 48. Network stationcomprising an apparatus according to claim
 24. 49. Computer-readablephysical storage medium embodying instructions configured, when executedon a computer processing unit, to carry out the method of claim 1.